[{"data":1,"prerenderedAt":2911},["ShallowReactive",2],{"global-content-nav":3,"raw-language":19,"main-content":148,"aside-major-pages":2806,"book-content-nav":2808,"quote-ids":2896},{"bookShortIds":4,"frontNavItems":6},[5],"foundations",[7,13],{"type":8,"shortId":9,"title":10,"flags":11,"link":12},"page","faq","Об \"Открытой математике\"",{},"\u002Fpage\u002Ffaq\u002F",{"type":14,"shortId":5,"title":15,"flags":16,"link":17,"children":18},"book","Основы математики",{},"\u002Fbook\u002Ffoundations\u002F",[],{"phrases":20,"functions":146},{"book":21,"content_nav":24,"x_contributors":26,"theme":29,"page":31,"pages":33,"x_sponsors":35,"theme_light":37,"group":39,"search":41,"materials":43,"key_elements":45,"stats":47,"theme_dark":49,"topic":51,"languages":52,"theme_system":54,"article":56,"settings":58,"built":60,"summary":62,"content":64,"practice":66,"erudit":68,"preview_content_page_description":70,"add_translation":72,"main_page":74,"contributors":76,"sponsors":78,"search_the_site":80,"searching_more":82,"no_results":84,"no_more_results":86,"no_content":88,"to_index":90,"about_textbook":92,"flag_dev":94,"flag_advanced":96,"flag_secondary":98,"contributors_description":100,"contributors_invite":102,"sponsors_description":104,"become_contributor":106,"become_sponsor":108,"no_contributors":110,"no_sponsors":112,"contribution":114,"editor":116,"contributor_page_description":118,"breadcrumb":120,"begin_learning":122,"connections":124,"updated":126,"externals_own":128,"externals_from":130,"need_to_know":132,"depends_on":134,"used_by":136,"externals":138,"article_seo_description":140,"summary_seo_description":142,"practice_seo_description":144},{"type":22,"value":23},"string","Учебник",{"type":22,"value":25},"Оглавление",{"type":27,"value":28},"function","(count) => m(count, \"Автор\", \"Автора\", \"Авторов\")",{"type":22,"value":30},"Тема",{"type":22,"value":32},"Страница",{"type":22,"value":34},"Страницы",{"type":27,"value":36},"(count) => m(count, \"Спонсор\", \"Спонсора\", \"Спонсоров\")",{"type":22,"value":38},"Светлая",{"type":22,"value":40},"Категория",{"type":22,"value":42},"Поиск",{"type":22,"value":44},"Материалы",{"type":22,"value":46},"Ключевые элементы",{"type":22,"value":48},"Статистика",{"type":22,"value":50},"Темная",{"type":22,"value":30},{"type":22,"value":53},"Языки",{"type":22,"value":55},"Системная",{"type":22,"value":57},"Статья",{"type":22,"value":59},"Настройки",{"type":22,"value":61},"Собрано",{"type":22,"value":63},"Конспект",{"type":22,"value":65},"Контент",{"type":22,"value":67},"Задачи",{"type":22,"value":69},"Erudit",{"type":22,"value":71},"Вы собираетесь перейти на страницу контента:",{"type":22,"value":73},"Добавить перевод",{"type":22,"value":75},"Главная страница",{"type":22,"value":77},"Авторы",{"type":22,"value":79},"Спонсоры",{"type":22,"value":81},"Поиск по сайту...",{"type":22,"value":83},"Ищем еще...",{"type":22,"value":85},"Ничего не найдено",{"type":22,"value":87},"Больше результатов нет",{"type":22,"value":89},"Контента нет.",{"type":22,"value":91},"К оглавлению",{"type":22,"value":93},"Об учебнике",{"type":22,"value":95},"Разработка",{"type":22,"value":97},"Профиль",{"type":22,"value":99},"Дополнение",{"type":22,"value":101},"Список людей, которые внесли вклад в материалы проекта: предложили ценные идеи, вносили корректировки в существующий материал или же написали собственный!",{"type":22,"value":103},"Вы тоже можете помочь проекту, внести свой вклад и попасть на эту страницу!",{"type":22,"value":105},"Список людей и организаций, которые поддерживают проект финансово. Благодаря им проект может существовать и развиваться. Если вы хотите помочь проекту, то тоже можете стать спонсором и попасть на эту страницу!",{"type":22,"value":107},"Стать автором",{"type":22,"value":109},"Стать спонсором",{"type":22,"value":111},"Пока что авторов нет. Вы можете стать первым!",{"type":22,"value":113},"Пока что спонсоров нет. Вы можете стать первым!",{"type":22,"value":115},"Вклад",{"type":22,"value":117},"Редактор",{"type":27,"value":119},"(name) => `Персональная страница автора \"${name}\" с данными о вкладе в проект.`",{"type":22,"value":121},"Путь",{"type":22,"value":123},"Начать изучение!",{"type":22,"value":125},"Связи",{"type":22,"value":127},"Обновлено",{"type":22,"value":129},"Собственные",{"type":22,"value":131},"Из",{"type":22,"value":133},"Надо знать",{"type":22,"value":135},"Зависит от",{"type":22,"value":137},"Используется в",{"type":22,"value":139},"Внешние",{"type":27,"value":141},"(contentTitle) => `Понятное и интересное объяснение темы «${contentTitle}». Показательные примеры, важные свойства, интересные факты, применение в жизни, понятные доказательства. Здесь вы точно разберетесь!`",{"type":27,"value":143},"(contentTitle) => `Конспект темы «${contentTitle}»: ключевые определения, теоремы, свойства и примеры их использования. Все самое важное и в кратком виде!`",{"type":27,"value":145},"(contentTitle) => `Разнообразные задачи с подсказками и ответами по теме «${contentTitle}». Интересные условия, подсказки и подробные решения. Превратите знания в навык!`",{"m":147},"function m(number, one, two, five, includeNumber = true) {\n  const text = [five, one, two, two, two, five][number % 100 > 10 && number % 100 \u003C 20 ? 0 : Math.min(number % 10, 5)];\n  return includeNumber ? `${number} ${text}` : text;\n}",{"contentPath":149,"mainContent":150},"summary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F",{"fullId":151,"shortId":151,"contentRelativePath":152,"title":153,"bookTitle":15,"breadcrumbs":154,"lastmod":161,"description":162,"decoration":163,"stats":164,"connections":170,"contributions":202,"type":174,"part":208,"parts":209,"prose":212,"storage":2699,"snippets":2719,"toc":2801},"foundations\u002Fpolynomials\u002Fspecial-products","01-foundations\u002F01-polynomials\u002F01-special-products","Формулы сокращенного умножения",[155,157],{"icon":156,"title":15,"link":17},"book-outline",{"icon":158,"title":159,"link":160},"folder-open","Многочлены","\u002Fgroup\u002Ffoundations\u002Fpolynomials\u002F","2026-05-01T18:59:52+03:00","\n    Квадрат суммы и разности, разность квадратов, куб суммы и разности -- очень полезные формулы, которые позволяют быстро раскладывать или запаковывать выражения со степенями (квадратами, кубами).\n  ","content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fdecoration.svg",{"elements":165},{"accent_term":166,"accent_statement":167,"accent_important":168,"problem":169},1,5,2,28,{"dependents":171,"externals":187},[172],{"type":173,"contentType":174,"title":175,"link":176,"uniques":177},"auto","topic","Выделение полного квадрата","\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square\u002F",[178,183],{"name":179,"title":180,"link":181,"schemaName":182},"squareSum","Формула квадрата суммы","\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=square-sum","accent_statement",{"name":184,"title":185,"link":186,"schemaName":182},"squareDiff","Формула квадрата разности","\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=square-diff",[188],{"type":189,"title":15,"items":190},"parent",[191,197],{"type":192,"title":193,"info":194,"link":195,"reason":196},"web","LibreTexts Mathematics","Открытая и бесплатная онлайн-коллекция учебных материалов по математике.","https:\u002F\u002Fmath.libretexts.org\u002F","Большое количество материалов и много типовых задач на отработку.",{"type":198,"title":199,"info":200,"reason":201},"physical","Математика. 6 класс","Мерзляк А.Г., Полонский В.Б., Якир М.С.","Огромное количество самых разных задач, от элементарнейших до довольно хитрых.",[203],{"contributorId":204,"name":205,"avatarUrl":206,"description":207},"gwynerva","Пётр Радько","\u002Ffile\u002Fcontributors\u002Fgwynerva\u002Favatar.png","Идея и основной контент.","summary",[210,208,211],"article","practice",{"__TSPROSE_proseElement":213,"schema":214,"children":218},true,{"name":215,"type":216,"linkable":217,"__TSPROSE_schema":213},"mix","block",false,[219,375,2461,2464,2483,2485],{"__TSPROSE_proseElement":213,"schema":220,"children":222,"id":374},{"name":221,"type":216,"linkable":213,"__TSPROSE_schema":213},"table",[223,244,260,295,327,342],{"__TSPROSE_proseElement":213,"schema":224,"children":227},{"name":225,"type":226,"linkable":217,"__TSPROSE_schema":213},"tableRow","inliner",[228],{"__TSPROSE_proseElement":213,"schema":229,"data":231,"children":233},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},"tableData",{"center":213,"hspan":232},3,[234],{"__TSPROSE_proseElement":213,"schema":235,"data":237,"children":239,"id":243},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},"emphasis",{"type":238},"bold",[240],{"__TSPROSE_proseElement":213,"schema":241,"data":153},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"text","emphasis-xDIirZKEZ",{"__TSPROSE_proseElement":213,"schema":245,"children":246},{"name":225,"type":226,"linkable":217,"__TSPROSE_schema":213},[247],{"__TSPROSE_proseElement":213,"schema":248,"data":249,"children":250},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},{"center":213,"hspan":232},[251],{"__TSPROSE_proseElement":213,"schema":252,"data":253,"children":255,"id":259},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":254},"italic",[256],{"__TSPROSE_proseElement":213,"schema":257,"data":258},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Квадраты -- Знать наизусть","emphasis-xdAI1B4uD",{"__TSPROSE_proseElement":213,"schema":261,"children":262},{"name":225,"type":226,"linkable":217,"__TSPROSE_schema":213},[263,274,285],{"__TSPROSE_proseElement":213,"schema":264,"children":265},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},[266],{"__TSPROSE_proseElement":213,"schema":267,"data":268,"children":269,"id":273},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[270],{"__TSPROSE_proseElement":213,"schema":271,"data":272},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Квадрат суммы и разности","emphasis-UgxvYeTh4",{"__TSPROSE_proseElement":213,"schema":275,"data":276,"children":277},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},{"center":213,"freeze":213},[278],{"__TSPROSE_proseElement":213,"schema":279,"data":281,"storageKey":283,"id":284},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},"inlinerMath",{"katex":282,"currentColor":213},"(a \\pm b)^2 = a^2 \\pm 2ab + b^2","$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 $","inlinerMath-3p8I79oez",{"__TSPROSE_proseElement":213,"schema":286,"children":287},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},[288],{"__TSPROSE_proseElement":213,"schema":289,"data":291,"storageKey":293,"id":294},{"name":290,"type":226,"linkable":213,"__TSPROSE_schema":213},"referenceInliner",{"label":292},"Примеры","\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fsummary\u002F$squareSumDiffDetails","referenceInliner-YYNdwYJxA",{"__TSPROSE_proseElement":213,"schema":296,"children":297},{"name":225,"type":226,"linkable":217,"__TSPROSE_schema":213},[298,309,319],{"__TSPROSE_proseElement":213,"schema":299,"children":300},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},[301],{"__TSPROSE_proseElement":213,"schema":302,"data":303,"children":304,"id":308},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[305],{"__TSPROSE_proseElement":213,"schema":306,"data":307},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Разность квадратов","emphasis-ZPAeZnQsd",{"__TSPROSE_proseElement":213,"schema":310,"data":311,"children":312},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},{"center":213,"freeze":213},[313],{"__TSPROSE_proseElement":213,"schema":314,"data":315,"storageKey":317,"id":318},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":316,"currentColor":213},"a^2 - b^2 = (a+b)(a-b)","$ a^2 - b^2 = (a+b)(a-b) $","inlinerMath-SoHR0k5lK",{"__TSPROSE_proseElement":213,"schema":320,"children":321},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},[322],{"__TSPROSE_proseElement":213,"schema":323,"data":324,"storageKey":325,"id":326},{"name":290,"type":226,"linkable":213,"__TSPROSE_schema":213},{"label":292},"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$diffOfSquaresExamples","referenceInliner-YYNdwYJxA-1",{"__TSPROSE_proseElement":213,"schema":328,"children":329},{"name":225,"type":226,"linkable":217,"__TSPROSE_schema":213},[330],{"__TSPROSE_proseElement":213,"schema":331,"data":332,"children":333},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},{"center":213,"hspan":232},[334],{"__TSPROSE_proseElement":213,"schema":335,"data":336,"children":337,"id":341},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":254},[338],{"__TSPROSE_proseElement":213,"schema":339,"data":340},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Кубы -- Знать не наизусть, но уметь распознать","emphasis-A2ddMKjfW",{"__TSPROSE_proseElement":213,"schema":343,"children":344},{"name":225,"type":226,"linkable":217,"__TSPROSE_schema":213},[345,356,366],{"__TSPROSE_proseElement":213,"schema":346,"children":347},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},[348],{"__TSPROSE_proseElement":213,"schema":349,"data":350,"children":351,"id":355},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[352],{"__TSPROSE_proseElement":213,"schema":353,"data":354},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Куб суммы и разности","emphasis-8y5FAMd3u",{"__TSPROSE_proseElement":213,"schema":357,"data":358,"children":359},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},{"center":213,"freeze":213},[360],{"__TSPROSE_proseElement":213,"schema":361,"data":362,"storageKey":364,"id":365},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":363,"currentColor":213},"(a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3","$ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $","inlinerMath-yqTRmYVBM",{"__TSPROSE_proseElement":213,"schema":367,"children":368},{"name":230,"type":226,"linkable":217,"__TSPROSE_schema":213},[369],{"__TSPROSE_proseElement":213,"schema":370,"data":371,"storageKey":372,"id":373},{"name":290,"type":226,"linkable":213,"__TSPROSE_schema":213},{"label":292},"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$cubeSumDiffExamples","referenceInliner-YYNdwYJxA-2","special-products-table",{"__TSPROSE_proseElement":213,"schema":376,"data":379,"children":381,"id":2460},{"name":377,"type":216,"linkable":378,"__TSPROSE_schema":213},"details","always",{"title":380},"Примеры квадрата суммы и разности",[382,799,1536],{"__TSPROSE_proseElement":213,"schema":383,"data":385,"children":389,"id":798},{"name":384,"type":216,"linkable":213,"__TSPROSE_schema":213},"problems",{"title":386,"level":387,"attributes":388},"Разложение по квадрату суммы","example",[],[390,398,497,582],{"__TSPROSE_proseElement":213,"schema":391,"children":393,"id":397},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},"paragraph",[394],{"__TSPROSE_proseElement":213,"schema":395,"data":396},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Раскройте скобки:","paragraph-ImJFceXZ8",{"__TSPROSE_proseElement":213,"schema":399,"data":401,"children":402},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},"subProblem",{},[403,414,422,431],{"__TSPROSE_proseElement":213,"schema":404,"children":406},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},"problemDescription",[407],{"__TSPROSE_proseElement":213,"schema":408,"data":410,"storageKey":412,"id":413},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},"blockMath",{"katex":411,"freeze":217},"(m+5)^2","$$ (m+5)^2 $$","blockMath-pV6CHNmGX",{"__TSPROSE_proseElement":213,"schema":415,"data":417},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},"problemCheck",{"serializedValidator":418},{"__ERUDIT_CHECK":213,"name":419,"data":420},"math-expression",{"expr":421},"m^2 + 10m + 25",{"__TSPROSE_proseElement":213,"schema":423,"children":425},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},"problemAnswer",[426],{"__TSPROSE_proseElement":213,"schema":427,"data":428,"storageKey":429,"id":430},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":421,"freeze":217},"$$ m^2 + 10m + 25 $$","blockMath-hDM8JJGyP",{"__TSPROSE_proseElement":213,"schema":432,"children":434},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},"problemSolution",[435,478,484,491],{"__TSPROSE_proseElement":213,"schema":436,"children":437,"id":477},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[438,441,447,450,456,459,465,468,474],{"__TSPROSE_proseElement":213,"schema":439,"data":440},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Роль ",{"__TSPROSE_proseElement":213,"schema":442,"data":443,"storageKey":445,"id":446},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":444},"a","$ a $","inlinerMath-YqDfaaRXA",{"__TSPROSE_proseElement":213,"schema":448,"data":449},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," играет буква ",{"__TSPROSE_proseElement":213,"schema":451,"data":452,"storageKey":454,"id":455},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":453},"m","$ m $","inlinerMath-uuvX6OFxQ",{"__TSPROSE_proseElement":213,"schema":457,"data":458},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},", а роль ",{"__TSPROSE_proseElement":213,"schema":460,"data":461,"storageKey":463,"id":464},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":462},"b","$ b $","inlinerMath-LTS8nNQCZ",{"__TSPROSE_proseElement":213,"schema":466,"data":467},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," число ",{"__TSPROSE_proseElement":213,"schema":469,"data":470,"storageKey":472,"id":473},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":471},"5","$ 5 $","inlinerMath-MAD3UaxRg",{"__TSPROSE_proseElement":213,"schema":475,"data":476},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Слева направо находим квадрат первого, удвоенное произведение первого на второе и квадрат второго. На таких простых числах делать это желательно в уме:","paragraph-HRg4ntc8Z",{"__TSPROSE_proseElement":213,"schema":479,"data":480,"storageKey":482,"id":483},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":481,"freeze":217},"m^2 >>{big} 2 \\cdot m \\cdot 5 = 10m >>{big} 5^2 = 25","$$ m^2 >>{big} 2 \\cdot m \\cdot 5 = 10m >>{big} 5^2 = 25 $$","blockMath-0KxHw2nNu",{"__TSPROSE_proseElement":213,"schema":485,"children":486,"id":490},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[487],{"__TSPROSE_proseElement":213,"schema":488,"data":489},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Записываем все эти числа через плюс и получаем готовое разложение:","paragraph-qkHzg2hJ6",{"__TSPROSE_proseElement":213,"schema":492,"data":493,"storageKey":495,"id":496},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":494,"freeze":217},"(m+5)^2 = m^2 + 10m + 25","$$ (m+5)^2 = m^2 + 10m + 25 $$","blockMath-6e3NAKPlk",{"__TSPROSE_proseElement":213,"schema":498,"data":499,"children":500},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[501,510,516,524],{"__TSPROSE_proseElement":213,"schema":502,"children":503},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[504],{"__TSPROSE_proseElement":213,"schema":505,"data":506,"storageKey":508,"id":509},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":507,"freeze":217},"\\left( 2 + \\frac{1}{8}x \\right)^2","$$ \\left( 2 + \\frac{1}{8}x \\right)^2 $$","blockMath-f7I6jVg0J",{"__TSPROSE_proseElement":213,"schema":511,"data":512},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":513},{"__ERUDIT_CHECK":213,"name":419,"data":514},{"expr":515},"4 + \\frac{1}{2}x + \\frac{1}{64}x^2",{"__TSPROSE_proseElement":213,"schema":517,"children":518},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[519],{"__TSPROSE_proseElement":213,"schema":520,"data":521,"storageKey":522,"id":523},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":515,"freeze":217},"$$ 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$","blockMath-L506HvSs3",{"__TSPROSE_proseElement":213,"schema":525,"children":526},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[527,564,570,576],{"__TSPROSE_proseElement":213,"schema":528,"children":529,"id":563},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[530,532,536,539,545,547,551,554,560],{"__TSPROSE_proseElement":213,"schema":531,"data":440},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":533,"data":534,"storageKey":445,"id":535},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":444},"inlinerMath-YqDfaaRXA-1",{"__TSPROSE_proseElement":213,"schema":537,"data":538},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," играет число ",{"__TSPROSE_proseElement":213,"schema":540,"data":541,"storageKey":543,"id":544},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":542},"2","$ 2 $","inlinerMath-z9Yar9waf",{"__TSPROSE_proseElement":213,"schema":546,"data":458},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":548,"data":549,"storageKey":463,"id":550},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":462},"inlinerMath-LTS8nNQCZ-1",{"__TSPROSE_proseElement":213,"schema":552,"data":553},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," дробь ",{"__TSPROSE_proseElement":213,"schema":555,"data":556,"storageKey":558,"id":559},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":557},"\\frac{1}{8}x","$ \\frac{1}{8}x $","inlinerMath-EYpHaQ5VW",{"__TSPROSE_proseElement":213,"schema":561,"data":562},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Слева направо находим квадрат первого, удвоенное произведение первого на второе и квадрат второго. Тут числа уже сложнее, поэтому можно считать все три по отдельности на бумаге. Не забывайте проводить сокращение!","paragraph-tRoi7xAAl",{"__TSPROSE_proseElement":213,"schema":565,"data":566,"storageKey":568,"id":569},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":567,"freeze":217},"2^2 = 4 >>{big} \\cancel{2} \\cdot \\cancel{2} \\cdot \\frac{1}{\\cancel{8}_{\\small\\cancel{4}_{\\small 2}}}x = \\frac{1}{2}x >>{big} \\left( \\frac{1}{8}x \\right)^2 = \\left(\\frac{1}{8}\\right)^2 x^2 = \\frac{1}{64}x^2","$$ 2^2 = 4 >>{big} \\cancel{2} \\cdot \\cancel{2} \\cdot \\frac{1}{\\cancel{8}_{\\small\\cancel{4}_{\\small 2}}}x = \\frac{1}{2}x >>{big} \\left( \\frac{1}{8}x \\right)^2 = \\left(\\frac{1}{8}\\right)^2 x^2 = \\frac{1}{64}x^2 $$","blockMath-UEXl8k1FG",{"__TSPROSE_proseElement":213,"schema":571,"children":572,"id":575},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[573],{"__TSPROSE_proseElement":213,"schema":574,"data":489},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-qkHzg2hJ6-1",{"__TSPROSE_proseElement":213,"schema":577,"data":578,"storageKey":580,"id":581},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":579,"freeze":217},"\\left( 2 + \\frac{1}{8}x \\right)^2 = 4 + \\frac{1}{2}x + \\frac{1}{64}x^2","$$ \\left( 2 + \\frac{1}{8}x \\right)^2 = 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$","blockMath-iW3hsY3J6",{"__TSPROSE_proseElement":213,"schema":583,"data":584,"children":585},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[586,595,601,634,642],{"__TSPROSE_proseElement":213,"schema":587,"children":588},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[589],{"__TSPROSE_proseElement":213,"schema":590,"data":591,"storageKey":593,"id":594},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":592,"freeze":217},"(-7a - 3b)^2","$$ (-7a - 3b)^2 $$","blockMath-FV48nxGyl",{"__TSPROSE_proseElement":213,"schema":596,"data":597},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":598},{"__ERUDIT_CHECK":213,"name":419,"data":599},{"expr":600},"49a^2 + 42ab + 9b^2",{"__TSPROSE_proseElement":213,"schema":602,"children":604},{"name":603,"type":216,"linkable":217,"__TSPROSE_schema":213},"problemHint",[605,621,627],{"__TSPROSE_proseElement":213,"schema":606,"children":607,"id":620},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[608,611,617],{"__TSPROSE_proseElement":213,"schema":609,"data":610},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Вариант ",{"__TSPROSE_proseElement":213,"schema":612,"data":613,"storageKey":615,"id":616},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":614},"1","$ 1 $","inlinerMath-GWJ3eIxvK",{"__TSPROSE_proseElement":213,"schema":618,"data":619},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," - запишите сумму под квадратом с явным знаком плюса:","paragraph-WUD9q4TRq",{"__TSPROSE_proseElement":213,"schema":622,"data":623,"storageKey":625,"id":626},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":624,"freeze":217},"(-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2","$$ (-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2 $$","blockMath-YyTXKOh9m",{"__TSPROSE_proseElement":213,"schema":628,"children":629,"id":633},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[630],{"__TSPROSE_proseElement":213,"schema":631,"data":632},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Вариант 2 -- внутри суммы вынесите минус за скобки.","paragraph-JvOAL3Gg5",{"__TSPROSE_proseElement":213,"schema":635,"children":636},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[637],{"__TSPROSE_proseElement":213,"schema":638,"data":639,"storageKey":640,"id":641},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":600,"freeze":217},"$$ 49a^2 + 42ab + 9b^2 $$","blockMath-JHMiXhAI7",{"__TSPROSE_proseElement":213,"schema":643,"children":644},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[645,761],{"__TSPROSE_proseElement":213,"schema":646,"data":648,"children":649},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},"problemSection","Через явный знак плюса",[650,706,710,743,749,755],{"__TSPROSE_proseElement":213,"schema":651,"children":652,"id":705},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[653,656,662,665,671,674,680,683,687,690,694,696,702],{"__TSPROSE_proseElement":213,"schema":654,"data":655},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Минус в сумме ",{"__TSPROSE_proseElement":213,"schema":657,"data":658,"storageKey":660,"id":661},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":659},"-7a - 3b","$ -7a - 3b $","inlinerMath-EOtl20nRc",{"__TSPROSE_proseElement":213,"schema":663,"data":664},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," можно рассматривать двояко. С одной стороны он может образовывать разность чисел ",{"__TSPROSE_proseElement":213,"schema":666,"data":667,"storageKey":669,"id":670},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":668},"-7a","$ -7a $","inlinerMath-80LHk8XHo",{"__TSPROSE_proseElement":213,"schema":672,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," и ",{"__TSPROSE_proseElement":213,"schema":675,"data":676,"storageKey":678,"id":679},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":677},"3b","$ 3b $","inlinerMath-OkRX6uD9W",{"__TSPROSE_proseElement":213,"schema":681,"data":682},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Но можно этот минус \"прилепить\" к ",{"__TSPROSE_proseElement":213,"schema":684,"data":685,"storageKey":678,"id":686},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":677},"inlinerMath-OkRX6uD9W-1",{"__TSPROSE_proseElement":213,"schema":688,"data":689},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". И тогда получится сумма двух чисел ",{"__TSPROSE_proseElement":213,"schema":691,"data":692,"storageKey":669,"id":693},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":668},"inlinerMath-80LHk8XHo-1",{"__TSPROSE_proseElement":213,"schema":695,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":697,"data":698,"storageKey":700,"id":701},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":699},"-3b","$ -3b $","inlinerMath-Djaufz4Sl",{"__TSPROSE_proseElement":213,"schema":703,"data":704},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},", которую явным образом можно записать вот так:","paragraph-WaBdeUTzI",{"__TSPROSE_proseElement":213,"schema":707,"data":708,"storageKey":625,"id":709},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":624,"freeze":217},"blockMath-YyTXKOh9m-1",{"__TSPROSE_proseElement":213,"schema":711,"children":712,"id":742},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[713,716,720,723,727,729,733,735,739],{"__TSPROSE_proseElement":213,"schema":714,"data":715},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"А дальше по уже отработанной схеме. Роль ",{"__TSPROSE_proseElement":213,"schema":717,"data":718,"storageKey":445,"id":719},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":444},"inlinerMath-YqDfaaRXA-2",{"__TSPROSE_proseElement":213,"schema":721,"data":722},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," в формуле квадрата суммы играет число ",{"__TSPROSE_proseElement":213,"schema":724,"data":725,"storageKey":669,"id":726},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":668},"inlinerMath-80LHk8XHo-2",{"__TSPROSE_proseElement":213,"schema":728,"data":458},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":730,"data":731,"storageKey":463,"id":732},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":462},"inlinerMath-LTS8nNQCZ-2",{"__TSPROSE_proseElement":213,"schema":734,"data":467},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":736,"data":737,"storageKey":700,"id":738},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":699},"inlinerMath-Djaufz4Sl-1",{"__TSPROSE_proseElement":213,"schema":740,"data":741},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Слева направо находим квадрат первого, удвоенное произведение первого на второе и квадрат второго.","paragraph-YgL94AVlu",{"__TSPROSE_proseElement":213,"schema":744,"data":745,"storageKey":747,"id":748},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":746,"freeze":217},"(-7a)^2 = (-7)^2 a^2 = 49a^2 >>{big} 2 \\cdot (-7a) \\cdot (-3b) = 42ab >>{big} (-3b)^2 = (-3)^2 b^2 = 9b^2","$$ (-7a)^2 = (-7)^2 a^2 = 49a^2 >>{big} 2 \\cdot (-7a) \\cdot (-3b) = 42ab >>{big} (-3b)^2 = (-3)^2 b^2 = 9b^2 $$","blockMath-r7gELRwOl",{"__TSPROSE_proseElement":213,"schema":750,"children":751,"id":754},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[752],{"__TSPROSE_proseElement":213,"schema":753,"data":489},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-qkHzg2hJ6-2",{"__TSPROSE_proseElement":213,"schema":756,"data":757,"storageKey":759,"id":760},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":758,"freeze":217},"(-7a - 3b)^2 = 49a^2 + 42ab + 9b^2","$$ (-7a - 3b)^2 = 49a^2 + 42ab + 9b^2 $$","blockMath-A2ZrMc5jo",{"__TSPROSE_proseElement":213,"schema":762,"data":763,"children":764},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},"Вынесение минуса за скобки",[765,779,785,792],{"__TSPROSE_proseElement":213,"schema":766,"children":767,"id":778},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[768,771,775],{"__TSPROSE_proseElement":213,"schema":769,"data":770},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Вынесем минус за скобки из выражения ",{"__TSPROSE_proseElement":213,"schema":772,"data":773,"storageKey":660,"id":774},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":659},"inlinerMath-EOtl20nRc-1",{"__TSPROSE_proseElement":213,"schema":776,"data":777},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},":","paragraph-8ANRciv58",{"__TSPROSE_proseElement":213,"schema":780,"data":781,"storageKey":783,"id":784},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":782,"freeze":217},"-7a - 3b = -(7a + 3b)","$$ -7a - 3b = -(7a + 3b) $$","blockMath-HfGK3huoD",{"__TSPROSE_proseElement":213,"schema":786,"children":787,"id":791},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[788],{"__TSPROSE_proseElement":213,"schema":789,"data":790},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Теперь у нас есть сумма внутри скобок, а отдельный минус \"уничтожит\" возведение в квадрат:","paragraph-6UJJ48xBQ",{"__TSPROSE_proseElement":213,"schema":793,"data":794,"storageKey":796,"id":797},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":795,"freeze":217},"(-(7a + 3b))^2 = (-1)^2 \\cdot (7a + 3b)^2 = (7a + 3b)^2 = 49a^2 + 42ab + 9b^2","$$ (-(7a + 3b))^2 = (-1)^2 \\cdot (7a + 3b)^2 = (7a + 3b)^2 = 49a^2 + 42ab + 9b^2 $$","blockMath-WkRLMEjNE","razlozhenie-po-kvadratu-summy",{"__TSPROSE_proseElement":213,"schema":800,"data":801,"children":804,"id":1535},{"name":384,"type":216,"linkable":213,"__TSPROSE_schema":213},{"title":802,"level":387,"attributes":803},"Запаковка в квадрат суммы",[],[805,812,981,1165,1340],{"__TSPROSE_proseElement":213,"schema":806,"children":807,"id":811},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[808],{"__TSPROSE_proseElement":213,"schema":809,"data":810},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Запишите сумму в виде квадрата суммы:","paragraph-aOnNwi11Y",{"__TSPROSE_proseElement":213,"schema":813,"data":814,"children":815},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[816,825,831,839],{"__TSPROSE_proseElement":213,"schema":817,"children":818},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[819],{"__TSPROSE_proseElement":213,"schema":820,"data":821,"storageKey":823,"id":824},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":822,"freeze":217},"49 + 14x + x^2","$$ 49 + 14x + x^2 $$","blockMath-lLCEZPE65",{"__TSPROSE_proseElement":213,"schema":826,"data":827},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":828},{"__ERUDIT_CHECK":213,"name":419,"data":829},{"expr":830},"(7 + x)^2",{"__TSPROSE_proseElement":213,"schema":832,"children":833},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[834],{"__TSPROSE_proseElement":213,"schema":835,"data":836,"storageKey":837,"id":838},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":830,"freeze":217},"$$ (7 + x)^2 $$","blockMath-k4z5iytxW",{"__TSPROSE_proseElement":213,"schema":840,"children":841},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[842,902],{"__TSPROSE_proseElement":213,"schema":843,"data":844,"children":845},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},"По крайним слагаемым",[846,896],{"__TSPROSE_proseElement":213,"schema":847,"children":848,"id":895},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[849,852,858,860,866,869,875,877,883,886,892],{"__TSPROSE_proseElement":213,"schema":850,"data":851},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Смотрим на крайние слагаемые: ",{"__TSPROSE_proseElement":213,"schema":853,"data":854,"storageKey":856,"id":857},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":855},"49 = 7^2","$ 49 = 7^2 $","inlinerMath-ocltGy9bI",{"__TSPROSE_proseElement":213,"schema":859,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":861,"data":862,"storageKey":864,"id":865},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":863},"x^2 = x^2","$ x^2 = x^2 $","inlinerMath-swnzwUffe",{"__TSPROSE_proseElement":213,"schema":867,"data":868},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Оба являются \"красивыми\" квадратами, значит сразу видно: ",{"__TSPROSE_proseElement":213,"schema":870,"data":871,"storageKey":873,"id":874},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":872},"a = 7","$ a = 7 $","inlinerMath-CiXrICT5m",{"__TSPROSE_proseElement":213,"schema":876,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":878,"data":879,"storageKey":881,"id":882},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":880},"b = x","$ b = x $","inlinerMath-wIHP4wnz6",{"__TSPROSE_proseElement":213,"schema":884,"data":885},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Проверяем центральное слагаемое: ",{"__TSPROSE_proseElement":213,"schema":887,"data":888,"storageKey":890,"id":891},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":889},"2ab = 2 \\cdot 7 \\cdot x = 14x","$ 2ab = 2 \\cdot 7 \\cdot x = 14x $","inlinerMath-aETNmkPHY",{"__TSPROSE_proseElement":213,"schema":893,"data":894},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," -- совпадает! Записываем результат:","paragraph-NtGwjlsYF",{"__TSPROSE_proseElement":213,"schema":897,"data":898,"storageKey":900,"id":901},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":899,"freeze":217},"49 + 14x + x^2 = \\underset{a^2}{7^2} + 2 \\cdot \\underset{a}{7} \\cdot \\underset{b}{x} + \\underset{b^2}{x^2} = (7 + x)^2","$$ 49 + 14x + x^2 = \\underset{a^2}{7^2} + 2 \\cdot \\underset{a}{7} \\cdot \\underset{b}{x} + \\underset{b^2}{x^2} = (7 + x)^2 $$","blockMath-eSZvu6RZE",{"__TSPROSE_proseElement":213,"schema":903,"data":904,"children":905},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},"По центральному слагаемому",[906,977],{"__TSPROSE_proseElement":213,"schema":907,"children":908,"id":976},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[909,912,918,921,925,928,934,937,943,945,951,954,960,963,967,969,973],{"__TSPROSE_proseElement":213,"schema":910,"data":911},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Начинаем с центрального слагаемого ",{"__TSPROSE_proseElement":213,"schema":913,"data":914,"storageKey":916,"id":917},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":915},"14x","$ 14x $","inlinerMath-wmtD0OrpQ",{"__TSPROSE_proseElement":213,"schema":919,"data":920},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Делим его на ",{"__TSPROSE_proseElement":213,"schema":922,"data":923,"storageKey":543,"id":924},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":542},"inlinerMath-z9Yar9waf-1",{"__TSPROSE_proseElement":213,"schema":926,"data":927},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," и получаем ",{"__TSPROSE_proseElement":213,"schema":929,"data":930,"storageKey":932,"id":933},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":931},"7x","$ 7x $","inlinerMath-ESNBsoN9g",{"__TSPROSE_proseElement":213,"schema":935,"data":936},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Это произведение надо разложить на два множителя так, чтобы их квадраты совпадали с крайними слагаемыми ",{"__TSPROSE_proseElement":213,"schema":938,"data":939,"storageKey":941,"id":942},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":940},"49","$ 49 $","inlinerMath-CKZbQKHFg",{"__TSPROSE_proseElement":213,"schema":944,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":946,"data":947,"storageKey":949,"id":950},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":948},"x^2","$ x^2 $","inlinerMath-tzIa3VWcL",{"__TSPROSE_proseElement":213,"schema":952,"data":953},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". В нашем случае делается это элементарно: ",{"__TSPROSE_proseElement":213,"schema":955,"data":956,"storageKey":958,"id":959},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":957},"7x = 7 \\cdot x","$ 7x = 7 \\cdot x $","inlinerMath-CCrBaCZ78",{"__TSPROSE_proseElement":213,"schema":961,"data":962},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},", значит ",{"__TSPROSE_proseElement":213,"schema":964,"data":965,"storageKey":873,"id":966},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":872},"inlinerMath-CiXrICT5m-1",{"__TSPROSE_proseElement":213,"schema":968,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":970,"data":971,"storageKey":881,"id":972},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":880},"inlinerMath-wIHP4wnz6-1",{"__TSPROSE_proseElement":213,"schema":974,"data":975},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},".","paragraph-tx26rhMWT",{"__TSPROSE_proseElement":213,"schema":978,"data":979,"storageKey":900,"id":980},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":899,"freeze":217},"blockMath-eSZvu6RZE-1",{"__TSPROSE_proseElement":213,"schema":982,"data":983,"children":984},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[985,994,1000,1016,1024],{"__TSPROSE_proseElement":213,"schema":986,"children":987},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[988],{"__TSPROSE_proseElement":213,"schema":989,"data":990,"storageKey":992,"id":993},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":991,"freeze":217},"1 + 8y + 16y^2","$$ 1 + 8y + 16y^2 $$","blockMath-xRWWHpy21",{"__TSPROSE_proseElement":213,"schema":995,"data":996},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":997},{"__ERUDIT_CHECK":213,"name":419,"data":998},{"expr":999},"(1 + 4y)^2",{"__TSPROSE_proseElement":213,"schema":1001,"children":1002},{"name":603,"type":216,"linkable":217,"__TSPROSE_schema":213},[1003],{"__TSPROSE_proseElement":213,"schema":1004,"children":1005,"id":1015},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1006,1009],{"__TSPROSE_proseElement":213,"schema":1007,"data":1008},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Воспользуйтесь тем, что ",{"__TSPROSE_proseElement":213,"schema":1010,"data":1011,"storageKey":1013,"id":1014},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1012},"1^2 = 1","$ 1^2 = 1 $","inlinerMath-ooHpGun5c","paragraph-JrwUzl6SV",{"__TSPROSE_proseElement":213,"schema":1017,"children":1018},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[1019],{"__TSPROSE_proseElement":213,"schema":1020,"data":1021,"storageKey":1022,"id":1023},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":999,"freeze":217},"$$ (1 + 4y)^2 $$","blockMath-Zb0IB7kUT",{"__TSPROSE_proseElement":213,"schema":1025,"children":1026},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[1027,1082],{"__TSPROSE_proseElement":213,"schema":1028,"data":844,"children":1029},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[1030,1076],{"__TSPROSE_proseElement":213,"schema":1031,"children":1032,"id":1075},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1033,1035,1041,1043,1049,1051,1057,1059,1065,1067,1073],{"__TSPROSE_proseElement":213,"schema":1034,"data":851},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1036,"data":1037,"storageKey":1039,"id":1040},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1038},"1 = 1^2","$ 1 = 1^2 $","inlinerMath-AM9HQ4pvg",{"__TSPROSE_proseElement":213,"schema":1042,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1044,"data":1045,"storageKey":1047,"id":1048},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1046},"16y^2 = (4y)^2","$ 16y^2 = (4y)^2 $","inlinerMath-AylrkEnVa",{"__TSPROSE_proseElement":213,"schema":1050,"data":868},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1052,"data":1053,"storageKey":1055,"id":1056},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1054},"a = 1","$ a = 1 $","inlinerMath-Wg9v6GZRc",{"__TSPROSE_proseElement":213,"schema":1058,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1060,"data":1061,"storageKey":1063,"id":1064},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1062},"b = 4y","$ b = 4y $","inlinerMath-xFYoFNmU1",{"__TSPROSE_proseElement":213,"schema":1066,"data":885},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1068,"data":1069,"storageKey":1071,"id":1072},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1070},"2ab = 2 \\cdot 1 \\cdot 4y = 8y","$ 2ab = 2 \\cdot 1 \\cdot 4y = 8y $","inlinerMath-7vUA7HQaF",{"__TSPROSE_proseElement":213,"schema":1074,"data":894},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-0wlPg6L16",{"__TSPROSE_proseElement":213,"schema":1077,"data":1078,"storageKey":1080,"id":1081},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1079,"freeze":217},"1 + 8y + 16y^2 = \\underset{a^2}{1^2} + 2 \\cdot \\underset{a}{1} \\cdot \\underset{b}{4y} + \\underset{b^2}{(4y)^2} = (1 + 4y)^2","$$ 1 + 8y + 16y^2 = \\underset{a^2}{1^2} + 2 \\cdot \\underset{a}{1} \\cdot \\underset{b}{4y} + \\underset{b^2}{(4y)^2} = (1 + 4y)^2 $$","blockMath-geppUWnXe",{"__TSPROSE_proseElement":213,"schema":1083,"data":904,"children":1084},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[1085,1161],{"__TSPROSE_proseElement":213,"schema":1086,"children":1087,"id":1160},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1088,1090,1096,1098,1102,1104,1110,1112,1116,1118,1124,1127,1131,1134,1140,1143,1149,1151,1157],{"__TSPROSE_proseElement":213,"schema":1089,"data":911},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1091,"data":1092,"storageKey":1094,"id":1095},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1093},"8y","$ 8y $","inlinerMath-eK5rwe15e",{"__TSPROSE_proseElement":213,"schema":1097,"data":920},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1099,"data":1100,"storageKey":543,"id":1101},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":542},"inlinerMath-z9Yar9waf-2",{"__TSPROSE_proseElement":213,"schema":1103,"data":927},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1105,"data":1106,"storageKey":1108,"id":1109},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1107},"4y","$ 4y $","inlinerMath-wQ95eiZjC",{"__TSPROSE_proseElement":213,"schema":1111,"data":936},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1113,"data":1114,"storageKey":615,"id":1115},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":614},"inlinerMath-GWJ3eIxvK-1",{"__TSPROSE_proseElement":213,"schema":1117,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1119,"data":1120,"storageKey":1122,"id":1123},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1121},"16y^2","$ 16y^2 $","inlinerMath-Z9kSxtG8j",{"__TSPROSE_proseElement":213,"schema":1125,"data":1126},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Вспоминаем, что ",{"__TSPROSE_proseElement":213,"schema":1128,"data":1129,"storageKey":1013,"id":1130},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1012},"inlinerMath-ooHpGun5c-1",{"__TSPROSE_proseElement":213,"schema":1132,"data":1133},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},", поэтому искусственно добавим единицу в разложение: ",{"__TSPROSE_proseElement":213,"schema":1135,"data":1136,"storageKey":1138,"id":1139},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1137},"1 \\cdot 4y","$ 1 \\cdot 4y $","inlinerMath-LxACR5myN",{"__TSPROSE_proseElement":213,"schema":1141,"data":1142},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Проверяем: ",{"__TSPROSE_proseElement":213,"schema":1144,"data":1145,"storageKey":1147,"id":1148},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1146},"1^2 = 1 = a","$ 1^2 = 1 = a $","inlinerMath-UKP1QET9S",{"__TSPROSE_proseElement":213,"schema":1150,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1152,"data":1153,"storageKey":1155,"id":1156},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1154},"(4y)^2 = 16y^2 = b^2","$ (4y)^2 = 16y^2 = b^2 $","inlinerMath-RPKOx7uC1",{"__TSPROSE_proseElement":213,"schema":1158,"data":1159},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Всё совпадает! Записываем квадрат суммы:","paragraph-XDwSVpSqD",{"__TSPROSE_proseElement":213,"schema":1162,"data":1163,"storageKey":1080,"id":1164},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1079,"freeze":217},"blockMath-geppUWnXe-1",{"__TSPROSE_proseElement":213,"schema":1166,"data":1167,"children":1168},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[1169,1178,1184,1194,1203],{"__TSPROSE_proseElement":213,"schema":1170,"children":1171},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[1172],{"__TSPROSE_proseElement":213,"schema":1173,"data":1174,"storageKey":1176,"id":1177},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1175,"freeze":217},"\\frac{1}{4}k^2 + k + 1","$$ \\frac{1}{4}k^2 + k + 1 $$","blockMath-9jIIVViSF",{"__TSPROSE_proseElement":213,"schema":1179,"data":1180},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":1181},{"__ERUDIT_CHECK":213,"name":419,"data":1182},{"expr":1183},"(k\u002F2 + 1)^2",{"__TSPROSE_proseElement":213,"schema":1185,"children":1186},{"name":603,"type":216,"linkable":217,"__TSPROSE_schema":213},[1187],{"__TSPROSE_proseElement":213,"schema":1188,"children":1189,"id":1193},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1190],{"__TSPROSE_proseElement":213,"schema":1191,"data":1192},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Не всегда центральное слагаемое нацело делится на два. Пускай вас это не останавливает...","paragraph-dbSQ5jiwT",{"__TSPROSE_proseElement":213,"schema":1195,"children":1196},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[1197],{"__TSPROSE_proseElement":213,"schema":1198,"data":1199,"storageKey":1201,"id":1202},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1200,"freeze":217},"\\left(\\frac{k}{2} + 1 \\right)^2","$$ \\left(\\frac{k}{2} + 1 \\right)^2 $$","blockMath-Sqh9IOrVG",{"__TSPROSE_proseElement":213,"schema":1204,"children":1205},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[1206,1259],{"__TSPROSE_proseElement":213,"schema":1207,"data":844,"children":1208},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[1209,1253],{"__TSPROSE_proseElement":213,"schema":1210,"children":1211,"id":1252},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1212,1214,1220,1222,1226,1228,1234,1236,1242,1244,1250],{"__TSPROSE_proseElement":213,"schema":1213,"data":851},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1215,"data":1216,"storageKey":1218,"id":1219},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1217},"\\frac{1}{4}k^2 = \\left(\\frac{k}{2}\\right)^2","$ \\frac{1}{4}k^2 = \\left(\\frac{k}{2}\\right)^2 $","inlinerMath-TfEAXhY2F",{"__TSPROSE_proseElement":213,"schema":1221,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1223,"data":1224,"storageKey":1039,"id":1225},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1038},"inlinerMath-AM9HQ4pvg-1",{"__TSPROSE_proseElement":213,"schema":1227,"data":868},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1229,"data":1230,"storageKey":1232,"id":1233},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1231},"a = \\frac{k}{2}","$ a = \\frac{k}{2} $","inlinerMath-EqVNcAV7k",{"__TSPROSE_proseElement":213,"schema":1235,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1237,"data":1238,"storageKey":1240,"id":1241},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1239},"b = 1","$ b = 1 $","inlinerMath-jT8WRtcC5",{"__TSPROSE_proseElement":213,"schema":1243,"data":885},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1245,"data":1246,"storageKey":1248,"id":1249},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1247},"2ab = 2 \\cdot \\frac{k}{2} \\cdot 1 = k","$ 2ab = 2 \\cdot \\frac{k}{2} \\cdot 1 = k $","inlinerMath-h10U39IqB",{"__TSPROSE_proseElement":213,"schema":1251,"data":894},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-r9kmXdc6h",{"__TSPROSE_proseElement":213,"schema":1254,"data":1255,"storageKey":1257,"id":1258},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1256,"freeze":217},"\\frac{1}{4}k^2 + k + 1 = \\underset{a^2}{\\left( \\frac{k}{2} \\right)^2} + 2 \\cdot \\underset{a}{\\frac{k}{2}} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = \\left(\\frac{k}{2} + 1 \\right)^2","$$ \\frac{1}{4}k^2 + k + 1 = \\underset{a^2}{\\left( \\frac{k}{2} \\right)^2} + 2 \\cdot \\underset{a}{\\frac{k}{2}} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = \\left(\\frac{k}{2} + 1 \\right)^2 $$","blockMath-OAtFOiXrE",{"__TSPROSE_proseElement":213,"schema":1260,"data":904,"children":1261},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[1262,1336],{"__TSPROSE_proseElement":213,"schema":1263,"children":1264,"id":1335},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1265,1267,1273,1275,1279,1282,1288,1291,1297,1299,1303,1305,1309,1311,1317,1319,1325,1327,1333],{"__TSPROSE_proseElement":213,"schema":1266,"data":911},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1268,"data":1269,"storageKey":1271,"id":1272},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1270},"k","$ k $","inlinerMath-KaVbgkHhg",{"__TSPROSE_proseElement":213,"schema":1274,"data":920},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1276,"data":1277,"storageKey":543,"id":1278},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":542},"inlinerMath-z9Yar9waf-3",{"__TSPROSE_proseElement":213,"schema":1280,"data":1281},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," даже несмотря на то, что оно не делится нацело. Получается ",{"__TSPROSE_proseElement":213,"schema":1283,"data":1284,"storageKey":1286,"id":1287},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1285},"\\frac{k}{2}","$ \\frac{k}{2} $","inlinerMath-IcB1YEh3y",{"__TSPROSE_proseElement":213,"schema":1289,"data":1290},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Нужно эту дробь запись через два множителя так, чтобы их квадраты совпадали с крайними слагаемыми ",{"__TSPROSE_proseElement":213,"schema":1292,"data":1293,"storageKey":1295,"id":1296},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1294},"\\frac{1}{4}k^2","$ \\frac{1}{4}k^2 $","inlinerMath-XhsYjlucP",{"__TSPROSE_proseElement":213,"schema":1298,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1300,"data":1301,"storageKey":615,"id":1302},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":614},"inlinerMath-GWJ3eIxvK-2",{"__TSPROSE_proseElement":213,"schema":1304,"data":1126},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1306,"data":1307,"storageKey":1013,"id":1308},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1012},"inlinerMath-ooHpGun5c-2",{"__TSPROSE_proseElement":213,"schema":1310,"data":1133},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1312,"data":1313,"storageKey":1315,"id":1316},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1314},"\\frac{k}{2} \\cdot 1","$ \\frac{k}{2} \\cdot 1 $","inlinerMath-5VcE9HQud",{"__TSPROSE_proseElement":213,"schema":1318,"data":1142},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1320,"data":1321,"storageKey":1323,"id":1324},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1322},"\\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2","$ \\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2 $","inlinerMath-9J027juAN",{"__TSPROSE_proseElement":213,"schema":1326,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1328,"data":1329,"storageKey":1331,"id":1332},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1330},"1^2 = 1 = b^2","$ 1^2 = 1 = b^2 $","inlinerMath-5XaGNX6Ix",{"__TSPROSE_proseElement":213,"schema":1334,"data":1159},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-eaXxIKRdy",{"__TSPROSE_proseElement":213,"schema":1337,"data":1338,"storageKey":1257,"id":1339},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1256,"freeze":217},"blockMath-OAtFOiXrE-1",{"__TSPROSE_proseElement":213,"schema":1341,"data":1342,"children":1343},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[1344,1353,1359,1369,1377],{"__TSPROSE_proseElement":213,"schema":1345,"children":1346},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[1347],{"__TSPROSE_proseElement":213,"schema":1348,"data":1349,"storageKey":1351,"id":1352},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1350,"freeze":217},"16t^2 + 36m^2 + 48tm","$$ 16t^2 + 36m^2 + 48tm $$","blockMath-BRO83bY8l",{"__TSPROSE_proseElement":213,"schema":1354,"data":1355},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":1356},{"__ERUDIT_CHECK":213,"name":419,"data":1357},{"expr":1358},"(4t + 6m)^2",{"__TSPROSE_proseElement":213,"schema":1360,"children":1361},{"name":603,"type":216,"linkable":217,"__TSPROSE_schema":213},[1362],{"__TSPROSE_proseElement":213,"schema":1363,"children":1364,"id":1368},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1365],{"__TSPROSE_proseElement":213,"schema":1366,"data":1367},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Не всегда слагаемые в сумме стоят в том же порядке, что и в формуле квадрата суммы. В таких случаях можно поменять их местами, пользуясь тем, что перестановка мест слагаемых сумму не меняет.","paragraph-CK7nCUn1u",{"__TSPROSE_proseElement":213,"schema":1370,"children":1371},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[1372],{"__TSPROSE_proseElement":213,"schema":1373,"data":1374,"storageKey":1375,"id":1376},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1358,"freeze":217},"$$ (4t + 6m)^2 $$","blockMath-YexRM6ll2",{"__TSPROSE_proseElement":213,"schema":1378,"children":1379},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[1380,1387,1393,1448],{"__TSPROSE_proseElement":213,"schema":1381,"children":1382,"id":1386},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1383],{"__TSPROSE_proseElement":213,"schema":1384,"data":1385},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"У суммы необычный вид: во втором слагаемом тоже есть квадрат. Поменяем местами второе и третье слагаемые (сумма от этого не поменяется), чтобы «чистые» квадраты оказались по краям, а смешанное слагаемое -- в середине:","paragraph-l9o0JtIer",{"__TSPROSE_proseElement":213,"schema":1388,"data":1389,"storageKey":1391,"id":1392},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1390,"freeze":217},"16t^2 + 48tm + 36m^2","$$ 16t^2 + 48tm + 36m^2 $$","blockMath-LJK09FYyD",{"__TSPROSE_proseElement":213,"schema":1394,"data":844,"children":1395},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[1396,1442],{"__TSPROSE_proseElement":213,"schema":1397,"children":1398,"id":1441},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1399,1401,1407,1409,1415,1417,1423,1425,1431,1433,1439],{"__TSPROSE_proseElement":213,"schema":1400,"data":851},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1402,"data":1403,"storageKey":1405,"id":1406},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1404},"16t^2 = (4t)^2","$ 16t^2 = (4t)^2 $","inlinerMath-qaVjaWTT2",{"__TSPROSE_proseElement":213,"schema":1408,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1410,"data":1411,"storageKey":1413,"id":1414},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1412},"36m^2 = (6m)^2","$ 36m^2 = (6m)^2 $","inlinerMath-44R58sl9K",{"__TSPROSE_proseElement":213,"schema":1416,"data":868},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1418,"data":1419,"storageKey":1421,"id":1422},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1420},"a = 4t","$ a = 4t $","inlinerMath-Dvo0ZB2yR",{"__TSPROSE_proseElement":213,"schema":1424,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1426,"data":1427,"storageKey":1429,"id":1430},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1428},"b = 6m","$ b = 6m $","inlinerMath-zfqINXAo1",{"__TSPROSE_proseElement":213,"schema":1432,"data":885},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1434,"data":1435,"storageKey":1437,"id":1438},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1436},"2ab = 2 \\cdot 4t \\cdot 6m = 48tm","$ 2ab = 2 \\cdot 4t \\cdot 6m = 48tm $","inlinerMath-ukNlMgNXk",{"__TSPROSE_proseElement":213,"schema":1440,"data":894},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-MGzj2Ab5u",{"__TSPROSE_proseElement":213,"schema":1443,"data":1444,"storageKey":1446,"id":1447},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1445,"freeze":217},"16t^2 + 36m^2 + 48tm = \\underset{a^2}{(4t)^2} + 2 \\cdot \\underset{a}{4t} \\cdot \\underset{b}{6m} + \\underset{b^2}{(6m)^2} = (4t + 6m)^2","$$ 16t^2 + 36m^2 + 48tm = \\underset{a^2}{(4t)^2} + 2 \\cdot \\underset{a}{4t} \\cdot \\underset{b}{6m} + \\underset{b^2}{(6m)^2} = (4t + 6m)^2 $$","blockMath-akHjm2BVw",{"__TSPROSE_proseElement":213,"schema":1449,"data":904,"children":1450},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[1451,1531],{"__TSPROSE_proseElement":213,"schema":1452,"children":1453,"id":1530},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1454,1456,1462,1464,1468,1470,1476,1478,1484,1486,1492,1494,1500,1502,1506,1508,1512,1514,1520,1522,1528],{"__TSPROSE_proseElement":213,"schema":1455,"data":911},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1457,"data":1458,"storageKey":1460,"id":1461},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1459},"48tm","$ 48tm $","inlinerMath-YwPHMKTdI",{"__TSPROSE_proseElement":213,"schema":1463,"data":920},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1465,"data":1466,"storageKey":543,"id":1467},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":542},"inlinerMath-z9Yar9waf-4",{"__TSPROSE_proseElement":213,"schema":1469,"data":927},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1471,"data":1472,"storageKey":1474,"id":1475},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1473},"24tm","$ 24tm $","inlinerMath-eWv1ACr1O",{"__TSPROSE_proseElement":213,"schema":1477,"data":936},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1479,"data":1480,"storageKey":1482,"id":1483},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1481},"16t^2","$ 16t^2 $","inlinerMath-QklJ2mV7I",{"__TSPROSE_proseElement":213,"schema":1485,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1487,"data":1488,"storageKey":1490,"id":1491},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1489},"36m^2","$ 36m^2 $","inlinerMath-DvGqpjAMd",{"__TSPROSE_proseElement":213,"schema":1493,"data":953},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1495,"data":1496,"storageKey":1498,"id":1499},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1497},"24tm = 4t \\cdot 6m","$ 24tm = 4t \\cdot 6m $","inlinerMath-OStviYHdm",{"__TSPROSE_proseElement":213,"schema":1501,"data":962},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1503,"data":1504,"storageKey":1421,"id":1505},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1420},"inlinerMath-Dvo0ZB2yR-1",{"__TSPROSE_proseElement":213,"schema":1507,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1509,"data":1510,"storageKey":1429,"id":1511},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1428},"inlinerMath-zfqINXAo1-1",{"__TSPROSE_proseElement":213,"schema":1513,"data":1142},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1515,"data":1516,"storageKey":1518,"id":1519},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1517},"(4t)^2 = 16t^2 = a^2","$ (4t)^2 = 16t^2 = a^2 $","inlinerMath-CGznEo7ze",{"__TSPROSE_proseElement":213,"schema":1521,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1523,"data":1524,"storageKey":1526,"id":1527},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1525},"(6m)^2 = 36m^2 = b^2","$ (6m)^2 = 36m^2 = b^2 $","inlinerMath-VR8AvPckN",{"__TSPROSE_proseElement":213,"schema":1529,"data":1159},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-DpwMVRKAJ",{"__TSPROSE_proseElement":213,"schema":1532,"data":1533,"storageKey":1446,"id":1534},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1445,"freeze":217},"blockMath-akHjm2BVw-1","zapakovka-v-kvadrat-summy",{"__TSPROSE_proseElement":213,"schema":1537,"data":1538,"children":1541,"id":2459},{"name":384,"type":216,"linkable":213,"__TSPROSE_schema":213},{"title":1539,"level":387,"attributes":1540},"Примеры использования квадрата разности",[],[1542,1674,1873,1987,2188,2278],{"__TSPROSE_proseElement":213,"schema":1543,"data":1544,"children":1545},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[1546,1561,1567,1575,1633],{"__TSPROSE_proseElement":213,"schema":1547,"children":1548},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[1549,1555],{"__TSPROSE_proseElement":213,"schema":1550,"children":1551,"id":1554},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1552],{"__TSPROSE_proseElement":213,"schema":1553,"data":396},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-ImJFceXZ8-1",{"__TSPROSE_proseElement":213,"schema":1556,"data":1557,"storageKey":1559,"id":1560},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1558,"freeze":217},"(6-c)^2","$$ (6-c)^2 $$","blockMath-ekLbCsNP4",{"__TSPROSE_proseElement":213,"schema":1562,"data":1563},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":1564},{"__ERUDIT_CHECK":213,"name":419,"data":1565},{"expr":1566},"36 - 12c + c^2",{"__TSPROSE_proseElement":213,"schema":1568,"children":1569},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[1570],{"__TSPROSE_proseElement":213,"schema":1571,"data":1572,"storageKey":1573,"id":1574},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1566,"freeze":217},"$$ 36 - 12c + c^2 $$","blockMath-LDyuZrOUH",{"__TSPROSE_proseElement":213,"schema":1576,"children":1577},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[1578,1614,1620,1627],{"__TSPROSE_proseElement":213,"schema":1579,"children":1580,"id":1613},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1581,1583,1587,1589,1595,1597,1601,1604,1610],{"__TSPROSE_proseElement":213,"schema":1582,"data":440},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1584,"data":1585,"storageKey":445,"id":1586},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":444},"inlinerMath-YqDfaaRXA-3",{"__TSPROSE_proseElement":213,"schema":1588,"data":538},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1590,"data":1591,"storageKey":1593,"id":1594},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1592},"6","$ 6 $","inlinerMath-9rSWhNw25",{"__TSPROSE_proseElement":213,"schema":1596,"data":458},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1598,"data":1599,"storageKey":463,"id":1600},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":462},"inlinerMath-LTS8nNQCZ-3",{"__TSPROSE_proseElement":213,"schema":1602,"data":1603},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," буква ",{"__TSPROSE_proseElement":213,"schema":1605,"data":1606,"storageKey":1608,"id":1609},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1607},"c","$ c $","inlinerMath-8Adf0APnu",{"__TSPROSE_proseElement":213,"schema":1611,"data":1612},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Слева направо находим квадрат первого, взятое с минусом удвоенное произведение первого на второе и квадрат второго. С такими простыми числами делать вычисления желательно в уме:","paragraph-Ec3dAgDra",{"__TSPROSE_proseElement":213,"schema":1615,"data":1616,"storageKey":1618,"id":1619},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1617,"freeze":217},"6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2","$$ 6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2 $$","blockMath-goZZUGvb0",{"__TSPROSE_proseElement":213,"schema":1621,"children":1622,"id":1626},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1623],{"__TSPROSE_proseElement":213,"schema":1624,"data":1625},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Записываем все полученные числа вместе:","paragraph-Y4rv4e5tI",{"__TSPROSE_proseElement":213,"schema":1628,"data":1629,"storageKey":1631,"id":1632},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1630,"freeze":217},"(6-c)^2 = 36 - 12c + c^2","$$ (6-c)^2 = 36 - 12c + c^2 $$","blockMath-1J8WJP6EN",{"__TSPROSE_proseElement":213,"schema":1634,"children":1636},{"name":1635,"type":216,"linkable":217,"__TSPROSE_schema":213},"problemNote",[1637,1653,1659],{"__TSPROSE_proseElement":213,"schema":1638,"children":1639,"id":1652},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1640,1643,1649],{"__TSPROSE_proseElement":213,"schema":1641,"data":1642},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Многие новички часто совершают ошибку \"двойного минуса\". При применении формулы квадрата разности один минус они берут \"из формулы\", а второй от ",{"__TSPROSE_proseElement":213,"schema":1644,"data":1645,"storageKey":1647,"id":1648},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1646},"-b","$ -b $","inlinerMath-MMRLAwV3s",{"__TSPROSE_proseElement":213,"schema":1650,"data":1651},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". В итоге минус на минус у них дает плюс и получается неправильный ответ:","paragraph-Im7JQUfFE",{"__TSPROSE_proseElement":213,"schema":1654,"data":1655,"storageKey":1657,"id":1658},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1656,"freeze":217},"(6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2}","$$ (6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2} $$","blockMath-ucfrketpu",{"__TSPROSE_proseElement":213,"schema":1660,"children":1661,"id":1673},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1662,1665],{"__TSPROSE_proseElement":213,"schema":1663,"data":1664},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Запомните раз и навсегда -- ",{"__TSPROSE_proseElement":213,"schema":1666,"data":1667,"children":1668,"id":1672},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[1669],{"__TSPROSE_proseElement":213,"schema":1670,"data":1671},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"любой минус уже \"учтен\" заранее в квадрате разности! Вообще не думайте про знаки!","emphasis-oQr58qZ9e","paragraph-ZHoY3bUYV",{"__TSPROSE_proseElement":213,"schema":1675,"data":1676,"children":1677},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[1678,1694,1700,1717,1725],{"__TSPROSE_proseElement":213,"schema":1679,"children":1680},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[1681,1688],{"__TSPROSE_proseElement":213,"schema":1682,"children":1683,"id":1687},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1684],{"__TSPROSE_proseElement":213,"schema":1685,"data":1686},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Запишите сумму в виде квадрата разности:","paragraph-gIPXco1hw",{"__TSPROSE_proseElement":213,"schema":1689,"data":1690,"storageKey":1692,"id":1693},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1691,"freeze":217},"9x^2 - 6x + 1","$$ 9x^2 - 6x + 1 $$","blockMath-1DwYljmox",{"__TSPROSE_proseElement":213,"schema":1695,"data":1696},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":1697},{"__ERUDIT_CHECK":213,"name":419,"data":1698},{"expr":1699},"(3x - 1)^2",{"__TSPROSE_proseElement":213,"schema":1701,"children":1702},{"name":603,"type":216,"linkable":217,"__TSPROSE_schema":213},[1703],{"__TSPROSE_proseElement":213,"schema":1704,"children":1705,"id":1716},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1706,1709,1713],{"__TSPROSE_proseElement":213,"schema":1707,"data":1708},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Вы ведь не забыли, что ",{"__TSPROSE_proseElement":213,"schema":1710,"data":1711,"storageKey":1013,"id":1712},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1012},"inlinerMath-ooHpGun5c-3",{"__TSPROSE_proseElement":213,"schema":1714,"data":1715},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"?","paragraph-KML9cUnPK",{"__TSPROSE_proseElement":213,"schema":1718,"children":1719},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[1720],{"__TSPROSE_proseElement":213,"schema":1721,"data":1722,"storageKey":1723,"id":1724},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1699,"freeze":217},"$$ (3x - 1)^2 $$","blockMath-p7CiNdIap",{"__TSPROSE_proseElement":213,"schema":1726,"children":1727},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[1728,1779],{"__TSPROSE_proseElement":213,"schema":1729,"data":844,"children":1730},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[1731,1773],{"__TSPROSE_proseElement":213,"schema":1732,"children":1733,"id":1772},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1734,1736,1742,1744,1748,1750,1756,1758,1762,1764,1770],{"__TSPROSE_proseElement":213,"schema":1735,"data":851},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1737,"data":1738,"storageKey":1740,"id":1741},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1739},"9x^2 = (3x)^2","$ 9x^2 = (3x)^2 $","inlinerMath-JHiC7V8Qr",{"__TSPROSE_proseElement":213,"schema":1743,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1745,"data":1746,"storageKey":1039,"id":1747},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1038},"inlinerMath-AM9HQ4pvg-2",{"__TSPROSE_proseElement":213,"schema":1749,"data":868},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1751,"data":1752,"storageKey":1754,"id":1755},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1753},"a = 3x","$ a = 3x $","inlinerMath-gOFjcUN5E",{"__TSPROSE_proseElement":213,"schema":1757,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1759,"data":1760,"storageKey":1240,"id":1761},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1239},"inlinerMath-jT8WRtcC5-1",{"__TSPROSE_proseElement":213,"schema":1763,"data":885},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1765,"data":1766,"storageKey":1768,"id":1769},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1767},"-2ab = -2 \\cdot 3x \\cdot 1 = -6x","$ -2ab = -2 \\cdot 3x \\cdot 1 = -6x $","inlinerMath-bd0EntIKb",{"__TSPROSE_proseElement":213,"schema":1771,"data":894},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-vdwWP5AUV",{"__TSPROSE_proseElement":213,"schema":1774,"data":1775,"storageKey":1777,"id":1778},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1776,"freeze":217},"9x^2 - 6x + 1 = \\underset{a^2}{(3x)^2} - 2 \\cdot \\underset{a}{3x} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = (3x - 1)^2","$$ 9x^2 - 6x + 1 = \\underset{a^2}{(3x)^2} - 2 \\cdot \\underset{a}{3x} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = (3x - 1)^2 $$","blockMath-y8LnumBRS",{"__TSPROSE_proseElement":213,"schema":1780,"data":904,"children":1781},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[1782,1869],{"__TSPROSE_proseElement":213,"schema":1783,"children":1784,"id":1868},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1785,1787,1793,1795,1801,1803,1809,1811,1817,1819,1823,1825,1829,1831,1837,1839,1843,1845,1849,1851,1857,1859,1865],{"__TSPROSE_proseElement":213,"schema":1786,"data":911},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1788,"data":1789,"storageKey":1791,"id":1792},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1790},"-6x","$ -6x $","inlinerMath-MG3Leuh7I",{"__TSPROSE_proseElement":213,"schema":1794,"data":920},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1796,"data":1797,"storageKey":1799,"id":1800},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1798},"-2","$ -2 $","inlinerMath-02bjIi1Ry",{"__TSPROSE_proseElement":213,"schema":1802,"data":927},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1804,"data":1805,"storageKey":1807,"id":1808},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1806},"3x","$ 3x $","inlinerMath-WYRLesvjE",{"__TSPROSE_proseElement":213,"schema":1810,"data":936},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1812,"data":1813,"storageKey":1815,"id":1816},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1814},"9x^2","$ 9x^2 $","inlinerMath-oeT1YYBtQ",{"__TSPROSE_proseElement":213,"schema":1818,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1820,"data":1821,"storageKey":615,"id":1822},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":614},"inlinerMath-GWJ3eIxvK-3",{"__TSPROSE_proseElement":213,"schema":1824,"data":1126},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1826,"data":1827,"storageKey":1013,"id":1828},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1012},"inlinerMath-ooHpGun5c-4",{"__TSPROSE_proseElement":213,"schema":1830,"data":1133},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1832,"data":1833,"storageKey":1835,"id":1836},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1834},"3x = 3x \\cdot 1","$ 3x = 3x \\cdot 1 $","inlinerMath-nbioH3US9",{"__TSPROSE_proseElement":213,"schema":1838,"data":962},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1840,"data":1841,"storageKey":1754,"id":1842},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1753},"inlinerMath-gOFjcUN5E-1",{"__TSPROSE_proseElement":213,"schema":1844,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1846,"data":1847,"storageKey":1240,"id":1848},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1239},"inlinerMath-jT8WRtcC5-2",{"__TSPROSE_proseElement":213,"schema":1850,"data":1142},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1852,"data":1853,"storageKey":1855,"id":1856},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1854},"9x^2 = (3x)^2 = a^2","$ 9x^2 = (3x)^2 = a^2 $","inlinerMath-BPY4xTos9",{"__TSPROSE_proseElement":213,"schema":1858,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1860,"data":1861,"storageKey":1863,"id":1864},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1862},"1 = 1^2 = b^2","$ 1 = 1^2 = b^2 $","inlinerMath-BRasBzsqn",{"__TSPROSE_proseElement":213,"schema":1866,"data":1867},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Всё совпадает! Записываем квадрат разности:","paragraph-LzAmB3G8J",{"__TSPROSE_proseElement":213,"schema":1870,"data":1871,"storageKey":1777,"id":1872},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1776,"freeze":217},"blockMath-y8LnumBRS-1",{"__TSPROSE_proseElement":213,"schema":1874,"data":1875,"children":1876},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[1877,1892,1898,1908,1916],{"__TSPROSE_proseElement":213,"schema":1878,"children":1879},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[1880,1886],{"__TSPROSE_proseElement":213,"schema":1881,"children":1882,"id":1885},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1883],{"__TSPROSE_proseElement":213,"schema":1884,"data":396},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-ImJFceXZ8-2",{"__TSPROSE_proseElement":213,"schema":1887,"data":1888,"storageKey":1890,"id":1891},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1889,"freeze":217},"(-7 + 2a)^2","$$ (-7 + 2a)^2 $$","blockMath-UAZ3MmPrY",{"__TSPROSE_proseElement":213,"schema":1893,"data":1894},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":1895},{"__ERUDIT_CHECK":213,"name":419,"data":1896},{"expr":1897},"4a^2 - 28a + 49",{"__TSPROSE_proseElement":213,"schema":1899,"children":1900},{"name":603,"type":216,"linkable":217,"__TSPROSE_schema":213},[1901],{"__TSPROSE_proseElement":213,"schema":1902,"children":1903,"id":1907},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1904],{"__TSPROSE_proseElement":213,"schema":1905,"data":1906},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Воспользуйтесь тем, что при перестановке мест слагаемых сумма не меняется.","paragraph-VnK6NjAgx",{"__TSPROSE_proseElement":213,"schema":1909,"children":1910},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[1911],{"__TSPROSE_proseElement":213,"schema":1912,"data":1913,"storageKey":1914,"id":1915},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1897,"freeze":217},"$$ 4a^2 - 28a + 49 $$","blockMath-UKdlYItNi",{"__TSPROSE_proseElement":213,"schema":1917,"children":1918},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[1919,1926,1932,1969,1975,1981],{"__TSPROSE_proseElement":213,"schema":1920,"children":1921,"id":1925},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1922],{"__TSPROSE_proseElement":213,"schema":1923,"data":1924},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"У суммы необычный вид. Минус стоит не в середине, а в начале. Поменяем местами первое и второе слагаемые (сумма от этого не поменяется), чтобы было удобнее сравнивать с формулой квадрата разности:","paragraph-bFAil9kuv",{"__TSPROSE_proseElement":213,"schema":1927,"data":1928,"storageKey":1930,"id":1931},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1929,"freeze":217},"(-7 + 2a)^2 = (2a - 7)^2","$$ (-7 + 2a)^2 = (2a - 7)^2 $$","blockMath-vPY8ftQWb",{"__TSPROSE_proseElement":213,"schema":1933,"children":1934,"id":1968},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1935,1938,1942,1945,1951,1953,1957,1959,1965],{"__TSPROSE_proseElement":213,"schema":1936,"data":1937},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Теперь роль ",{"__TSPROSE_proseElement":213,"schema":1939,"data":1940,"storageKey":445,"id":1941},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":444},"inlinerMath-YqDfaaRXA-4",{"__TSPROSE_proseElement":213,"schema":1943,"data":1944},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," в формуле квадрата разности играет число ",{"__TSPROSE_proseElement":213,"schema":1946,"data":1947,"storageKey":1949,"id":1950},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1948},"2a","$ 2a $","inlinerMath-uofr24ZDY",{"__TSPROSE_proseElement":213,"schema":1952,"data":458},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1954,"data":1955,"storageKey":463,"id":1956},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":462},"inlinerMath-LTS8nNQCZ-4",{"__TSPROSE_proseElement":213,"schema":1958,"data":1603},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":1960,"data":1961,"storageKey":1963,"id":1964},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1962},"7","$ 7 $","inlinerMath-sYh3uEvdQ",{"__TSPROSE_proseElement":213,"schema":1966,"data":1967},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Слева направо находим квадрат первого, взятое с минусом удвоенное произведение первого на второе и квадрат второго:","paragraph-6m1tMiHl0",{"__TSPROSE_proseElement":213,"schema":1970,"data":1971,"storageKey":1973,"id":1974},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1972,"freeze":217},"(2a)^2 = 4a^2 >>{big} -2 \\cdot 2a \\cdot 7 = -28a >>{big} 7^2 = 49","$$ (2a)^2 = 4a^2 >>{big} -2 \\cdot 2a \\cdot 7 = -28a >>{big} 7^2 = 49 $$","blockMath-iIPFwePt6",{"__TSPROSE_proseElement":213,"schema":1976,"children":1977,"id":1980},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1978],{"__TSPROSE_proseElement":213,"schema":1979,"data":1625},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-Y4rv4e5tI-1",{"__TSPROSE_proseElement":213,"schema":1982,"data":1983,"storageKey":1985,"id":1986},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":1984,"freeze":217},"(-7 + 2a)^2 = 4a^2 - 28a + 49","$$ (-7 + 2a)^2 = 4a^2 - 28a + 49 $$","blockMath-CQRD6g7ty",{"__TSPROSE_proseElement":213,"schema":1988,"data":1989,"children":1990},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[1991,2006,2012,2022,2030],{"__TSPROSE_proseElement":213,"schema":1992,"children":1993},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[1994,2000],{"__TSPROSE_proseElement":213,"schema":1995,"children":1996,"id":1999},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[1997],{"__TSPROSE_proseElement":213,"schema":1998,"data":1686},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-gIPXco1hw-1",{"__TSPROSE_proseElement":213,"schema":2001,"data":2002,"storageKey":2004,"id":2005},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2003,"freeze":217},"- 12k + 4k^2 + 9","$$ - 12k + 4k^2 + 9 $$","blockMath-6wHlw0P5m",{"__TSPROSE_proseElement":213,"schema":2007,"data":2008},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":2009},{"__ERUDIT_CHECK":213,"name":419,"data":2010},{"expr":2011},"(2k - 3)^2",{"__TSPROSE_proseElement":213,"schema":2013,"children":2014},{"name":603,"type":216,"linkable":217,"__TSPROSE_schema":213},[2015],{"__TSPROSE_proseElement":213,"schema":2016,"children":2017,"id":2021},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2018],{"__TSPROSE_proseElement":213,"schema":2019,"data":2020},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Переставьте слагаемые в сумме в том же порядке, что и в формуле квадрата разности.","paragraph-tZomLVu6r",{"__TSPROSE_proseElement":213,"schema":2023,"children":2024},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[2025],{"__TSPROSE_proseElement":213,"schema":2026,"data":2027,"storageKey":2028,"id":2029},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2011,"freeze":217},"$$ (2k - 3)^2 $$","blockMath-pruShJQyn",{"__TSPROSE_proseElement":213,"schema":2031,"children":2032},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[2033,2040,2046,2101],{"__TSPROSE_proseElement":213,"schema":2034,"children":2035,"id":2039},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2036],{"__TSPROSE_proseElement":213,"schema":2037,"data":2038},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"У суммы необычный вид: «чистые» квадраты не стоят по краям. Переставим слагаемые (сумма от этого не поменяется), чтобы смешанное слагаемое оказалось в середине:","paragraph-CShLaETXc",{"__TSPROSE_proseElement":213,"schema":2041,"data":2042,"storageKey":2044,"id":2045},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2043,"freeze":217},"- 12k + 4k^2 + 9 = 4k^2 - 12k + 9","$$ - 12k + 4k^2 + 9 = 4k^2 - 12k + 9 $$","blockMath-DtE8x3QKN",{"__TSPROSE_proseElement":213,"schema":2047,"data":844,"children":2048},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[2049,2095],{"__TSPROSE_proseElement":213,"schema":2050,"children":2051,"id":2094},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2052,2054,2060,2062,2068,2070,2076,2078,2084,2086,2092],{"__TSPROSE_proseElement":213,"schema":2053,"data":851},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2055,"data":2056,"storageKey":2058,"id":2059},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2057},"4k^2 = (2k)^2","$ 4k^2 = (2k)^2 $","inlinerMath-JJuiBLiUp",{"__TSPROSE_proseElement":213,"schema":2061,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2063,"data":2064,"storageKey":2066,"id":2067},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2065},"9 = 3^2","$ 9 = 3^2 $","inlinerMath-WGLl6WNtY",{"__TSPROSE_proseElement":213,"schema":2069,"data":868},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2071,"data":2072,"storageKey":2074,"id":2075},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2073},"a = 2k","$ a = 2k $","inlinerMath-uOpB6qvj4",{"__TSPROSE_proseElement":213,"schema":2077,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2079,"data":2080,"storageKey":2082,"id":2083},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2081},"b = 3","$ b = 3 $","inlinerMath-FrcWrXeOp",{"__TSPROSE_proseElement":213,"schema":2085,"data":885},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2087,"data":2088,"storageKey":2090,"id":2091},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2089},"-2ab = -2 \\cdot 2k \\cdot 3 = -12k","$ -2ab = -2 \\cdot 2k \\cdot 3 = -12k $","inlinerMath-tpKMpxWgT",{"__TSPROSE_proseElement":213,"schema":2093,"data":894},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-SqTBOqrpk",{"__TSPROSE_proseElement":213,"schema":2096,"data":2097,"storageKey":2099,"id":2100},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2098,"freeze":217},"4k^2 - 12k + 9 = \\underset{a^2}{(2k)^2} - 2 \\cdot \\underset{a}{2k} \\cdot \\underset{b}{3} + \\underset{b^2}{3^2} = (2k - 3)^2","$$ 4k^2 - 12k + 9 = \\underset{a^2}{(2k)^2} - 2 \\cdot \\underset{a}{2k} \\cdot \\underset{b}{3} + \\underset{b^2}{3^2} = (2k - 3)^2 $$","blockMath-S0jbGcHVk",{"__TSPROSE_proseElement":213,"schema":2102,"data":904,"children":2103},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[2104,2184],{"__TSPROSE_proseElement":213,"schema":2105,"children":2106,"id":2183},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2107,2109,2115,2117,2121,2123,2129,2131,2137,2139,2145,2147,2153,2155,2159,2161,2165,2167,2173,2175,2181],{"__TSPROSE_proseElement":213,"schema":2108,"data":911},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2110,"data":2111,"storageKey":2113,"id":2114},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2112},"-12k","$ -12k $","inlinerMath-WgJ1mg3NQ",{"__TSPROSE_proseElement":213,"schema":2116,"data":920},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2118,"data":2119,"storageKey":1799,"id":2120},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1798},"inlinerMath-02bjIi1Ry-1",{"__TSPROSE_proseElement":213,"schema":2122,"data":927},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2124,"data":2125,"storageKey":2127,"id":2128},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2126},"6k","$ 6k $","inlinerMath-OSxZQEvnW",{"__TSPROSE_proseElement":213,"schema":2130,"data":936},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2132,"data":2133,"storageKey":2135,"id":2136},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2134},"4k^2","$ 4k^2 $","inlinerMath-OGjRamVXQ",{"__TSPROSE_proseElement":213,"schema":2138,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2140,"data":2141,"storageKey":2143,"id":2144},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2142},"9","$ 9 $","inlinerMath-Ss7zKstt6",{"__TSPROSE_proseElement":213,"schema":2146,"data":953},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2148,"data":2149,"storageKey":2151,"id":2152},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2150},"6k = 2k \\cdot 3","$ 6k = 2k \\cdot 3 $","inlinerMath-pRE4jTIIT",{"__TSPROSE_proseElement":213,"schema":2154,"data":962},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2156,"data":2157,"storageKey":2074,"id":2158},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2073},"inlinerMath-uOpB6qvj4-1",{"__TSPROSE_proseElement":213,"schema":2160,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2162,"data":2163,"storageKey":2082,"id":2164},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2081},"inlinerMath-FrcWrXeOp-1",{"__TSPROSE_proseElement":213,"schema":2166,"data":1142},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2168,"data":2169,"storageKey":2171,"id":2172},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2170},"(2k)^2 = 4k^2 = a^2","$ (2k)^2 = 4k^2 = a^2 $","inlinerMath-bjOiv16QN",{"__TSPROSE_proseElement":213,"schema":2174,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2176,"data":2177,"storageKey":2179,"id":2180},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2178},"3^2 = 9 = b^2","$ 3^2 = 9 = b^2 $","inlinerMath-xD4o3bMWT",{"__TSPROSE_proseElement":213,"schema":2182,"data":1867},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-zXy0VX4Gd",{"__TSPROSE_proseElement":213,"schema":2185,"data":2186,"storageKey":2099,"id":2187},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2098,"freeze":217},"blockMath-S0jbGcHVk-1",{"__TSPROSE_proseElement":213,"schema":2189,"data":2190,"children":2191},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[2192,2207,2213,2222],{"__TSPROSE_proseElement":213,"schema":2193,"children":2194},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[2195,2201],{"__TSPROSE_proseElement":213,"schema":2196,"children":2197,"id":2200},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2198],{"__TSPROSE_proseElement":213,"schema":2199,"data":396},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-ImJFceXZ8-3",{"__TSPROSE_proseElement":213,"schema":2202,"data":2203,"storageKey":2205,"id":2206},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2204,"freeze":217},"\\left( 5y - \\frac{3}{4}x \\right)^2","$$ \\left( 5y - \\frac{3}{4}x \\right)^2 $$","blockMath-vDseR9cWt",{"__TSPROSE_proseElement":213,"schema":2208,"data":2209},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":2210},{"__ERUDIT_CHECK":213,"name":419,"data":2211},{"expr":2212},"25y^2 - 15xy\u002F2 + 9x^2\u002F16",{"__TSPROSE_proseElement":213,"schema":2214,"children":2215},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[2216],{"__TSPROSE_proseElement":213,"schema":2217,"data":2218,"storageKey":2220,"id":2221},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2219,"freeze":217},"25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2","$$ 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$","blockMath-6qRF8kVXe",{"__TSPROSE_proseElement":213,"schema":2223,"children":2224},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[2225,2260,2266,2272],{"__TSPROSE_proseElement":213,"schema":2226,"children":2227,"id":2259},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2228,2230,2234,2236,2242,2244,2248,2250,2256],{"__TSPROSE_proseElement":213,"schema":2229,"data":440},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2231,"data":2232,"storageKey":445,"id":2233},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":444},"inlinerMath-YqDfaaRXA-5",{"__TSPROSE_proseElement":213,"schema":2235,"data":1944},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2237,"data":2238,"storageKey":2240,"id":2241},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2239},"5y","$ 5y $","inlinerMath-dfcEVhA4p",{"__TSPROSE_proseElement":213,"schema":2243,"data":458},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2245,"data":2246,"storageKey":463,"id":2247},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":462},"inlinerMath-LTS8nNQCZ-5",{"__TSPROSE_proseElement":213,"schema":2249,"data":1603},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2251,"data":2252,"storageKey":2254,"id":2255},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2253},"\\frac{3}{4}x","$ \\frac{3}{4}x $","inlinerMath-yaFjq8F3u",{"__TSPROSE_proseElement":213,"schema":2257,"data":2258},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Слева направо находим квадрат первого, взятое с минусом удвоенное произведение первого на второе и квадрат второго. Тут числа уже сложнее, поэтому можно считать все три по отдельности на бумаге. Не забывайте проводить сокращение!","paragraph-FTUibZKC1",{"__TSPROSE_proseElement":213,"schema":2261,"data":2262,"storageKey":2264,"id":2265},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2263,"freeze":217},"(5y)^2 = 5^2y^2 = 25y^2 >>{big} -\\cancel{2} \\cdot 5y \\cdot \\frac{3}{\\cancel{4}_{\\small 2}}x = -\\frac{15}{2}xy >>{big} \\left( \\frac{3}{4}x \\right)^2 = \\left(\\frac{3}{4}\\right)^2 x^2 = \\frac{9}{16}x^2","$$ (5y)^2 = 5^2y^2 = 25y^2 >>{big} -\\cancel{2} \\cdot 5y \\cdot \\frac{3}{\\cancel{4}_{\\small 2}}x = -\\frac{15}{2}xy >>{big} \\left( \\frac{3}{4}x \\right)^2 = \\left(\\frac{3}{4}\\right)^2 x^2 = \\frac{9}{16}x^2 $$","blockMath-ZNKwF9cMD",{"__TSPROSE_proseElement":213,"schema":2267,"children":2268,"id":2271},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2269],{"__TSPROSE_proseElement":213,"schema":2270,"data":1625},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-Y4rv4e5tI-2",{"__TSPROSE_proseElement":213,"schema":2273,"data":2274,"storageKey":2276,"id":2277},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2275,"freeze":217},"\\left( 5y - \\frac{3}{4}x \\right)^2 = 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2","$$ \\left( 5y - \\frac{3}{4}x \\right)^2 = 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$","blockMath-vJaS13MGL",{"__TSPROSE_proseElement":213,"schema":2279,"data":2280,"children":2281},{"name":400,"type":216,"linkable":217,"__TSPROSE_schema":213},{},[2282,2297,2303,2312],{"__TSPROSE_proseElement":213,"schema":2283,"children":2284},{"name":405,"type":216,"linkable":217,"__TSPROSE_schema":213},[2285,2291],{"__TSPROSE_proseElement":213,"schema":2286,"children":2287,"id":2290},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2288],{"__TSPROSE_proseElement":213,"schema":2289,"data":1686},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-gIPXco1hw-2",{"__TSPROSE_proseElement":213,"schema":2292,"data":2293,"storageKey":2295,"id":2296},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2294,"freeze":217},"\\frac{1}{4}m^2 - 5m + 25","$$ \\frac{1}{4}m^2 - 5m + 25 $$","blockMath-qErZGafTm",{"__TSPROSE_proseElement":213,"schema":2298,"data":2299},{"name":416,"type":216,"linkable":217,"__TSPROSE_schema":213},{"serializedValidator":2300},{"__ERUDIT_CHECK":213,"name":419,"data":2301},{"expr":2302},"(m\u002F2 - 5)^2",{"__TSPROSE_proseElement":213,"schema":2304,"children":2305},{"name":424,"type":216,"linkable":217,"__TSPROSE_schema":213},[2306],{"__TSPROSE_proseElement":213,"schema":2307,"data":2308,"storageKey":2310,"id":2311},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2309,"freeze":217},"\\left(\\frac{m}{2} - 5\\right)^2","$$ \\left(\\frac{m}{2} - 5\\right)^2 $$","blockMath-KELlMyhFW",{"__TSPROSE_proseElement":213,"schema":2313,"children":2314},{"name":433,"type":216,"linkable":217,"__TSPROSE_schema":213},[2315,2370],{"__TSPROSE_proseElement":213,"schema":2316,"data":844,"children":2317},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[2318,2364],{"__TSPROSE_proseElement":213,"schema":2319,"children":2320,"id":2363},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2321,2323,2329,2331,2337,2339,2345,2347,2353,2355,2361],{"__TSPROSE_proseElement":213,"schema":2322,"data":851},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2324,"data":2325,"storageKey":2327,"id":2328},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2326},"\\frac{1}{4}m^2 = \\left(\\frac{m}{2}\\right)^2","$ \\frac{1}{4}m^2 = \\left(\\frac{m}{2}\\right)^2 $","inlinerMath-tnO4zbQK3",{"__TSPROSE_proseElement":213,"schema":2330,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2332,"data":2333,"storageKey":2335,"id":2336},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2334},"25 = 5^2","$ 25 = 5^2 $","inlinerMath-CQjdOgpL8",{"__TSPROSE_proseElement":213,"schema":2338,"data":868},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2340,"data":2341,"storageKey":2343,"id":2344},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2342},"a = \\frac{m}{2}","$ a = \\frac{m}{2} $","inlinerMath-eUTZMMfdg",{"__TSPROSE_proseElement":213,"schema":2346,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2348,"data":2349,"storageKey":2351,"id":2352},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2350},"b = 5","$ b = 5 $","inlinerMath-pB2WH7ceV",{"__TSPROSE_proseElement":213,"schema":2354,"data":885},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2356,"data":2357,"storageKey":2359,"id":2360},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2358},"-2ab = -2 \\cdot \\frac{m}{2} \\cdot 5 = -5m","$ -2ab = -2 \\cdot \\frac{m}{2} \\cdot 5 = -5m $","inlinerMath-tFoidhqYH",{"__TSPROSE_proseElement":213,"schema":2362,"data":894},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-iazXiEpBC",{"__TSPROSE_proseElement":213,"schema":2365,"data":2366,"storageKey":2368,"id":2369},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2367,"freeze":217},"\\frac{1}{4}m^2 - 5m + 25 = \\underset{a^2}{\\left(\\frac{1}{2}m\\right)^2} - 2 \\cdot \\underset{a}{\\frac{1}{2}m} \\cdot \\underset{b}{5} + \\underset{b^2}{5^2} = \\left(\\frac{m}{2} - 5\\right)^2","$$ \\frac{1}{4}m^2 - 5m + 25 = \\underset{a^2}{\\left(\\frac{1}{2}m\\right)^2} - 2 \\cdot \\underset{a}{\\frac{1}{2}m} \\cdot \\underset{b}{5} + \\underset{b^2}{5^2} = \\left(\\frac{m}{2} - 5\\right)^2 $$","blockMath-xBGmZf40t",{"__TSPROSE_proseElement":213,"schema":2371,"data":904,"children":2372},{"name":647,"type":216,"linkable":217,"__TSPROSE_schema":213},[2373,2455],{"__TSPROSE_proseElement":213,"schema":2374,"children":2375,"id":2454},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2376,2378,2384,2386,2390,2392,2398,2401,2407,2409,2415,2418,2424,2426,2430,2432,2436,2438,2444,2446,2452],{"__TSPROSE_proseElement":213,"schema":2377,"data":911},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2379,"data":2380,"storageKey":2382,"id":2383},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2381},"-5m","$ -5m $","inlinerMath-Aa3jYyDxo",{"__TSPROSE_proseElement":213,"schema":2385,"data":920},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2387,"data":2388,"storageKey":1799,"id":2389},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":1798},"inlinerMath-02bjIi1Ry-2",{"__TSPROSE_proseElement":213,"schema":2391,"data":1281},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2393,"data":2394,"storageKey":2396,"id":2397},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2395},"\\frac{5m}{2}","$ \\frac{5m}{2} $","inlinerMath-l3IYVzMkX",{"__TSPROSE_proseElement":213,"schema":2399,"data":2400},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Нужно эту дробь записать через два множителя так, чтобы их квадраты совпадали с крайними слагаемыми ",{"__TSPROSE_proseElement":213,"schema":2402,"data":2403,"storageKey":2405,"id":2406},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2404},"\\frac{1}{4}m^2","$ \\frac{1}{4}m^2 $","inlinerMath-xhICl1aoZ",{"__TSPROSE_proseElement":213,"schema":2408,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2410,"data":2411,"storageKey":2413,"id":2414},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2412},"25","$ 25 $","inlinerMath-iCjF04534",{"__TSPROSE_proseElement":213,"schema":2416,"data":2417},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". Раскладываем: ",{"__TSPROSE_proseElement":213,"schema":2419,"data":2420,"storageKey":2422,"id":2423},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2421},"\\frac{5m}{2} = \\frac{m}{2} \\cdot 5","$ \\frac{5m}{2} = \\frac{m}{2} \\cdot 5 $","inlinerMath-qax14IJhI",{"__TSPROSE_proseElement":213,"schema":2425,"data":962},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2427,"data":2428,"storageKey":2343,"id":2429},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2342},"inlinerMath-eUTZMMfdg-1",{"__TSPROSE_proseElement":213,"schema":2431,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2433,"data":2434,"storageKey":2351,"id":2435},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2350},"inlinerMath-pB2WH7ceV-1",{"__TSPROSE_proseElement":213,"schema":2437,"data":1142},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2439,"data":2440,"storageKey":2442,"id":2443},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2441},"\\left(\\frac{m}{2}\\right)^2 = \\frac{1}{4}m^2 = a^2","$ \\left(\\frac{m}{2}\\right)^2 = \\frac{1}{4}m^2 = a^2 $","inlinerMath-3XGUv7kWj",{"__TSPROSE_proseElement":213,"schema":2445,"data":673},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2447,"data":2448,"storageKey":2450,"id":2451},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2449},"5^2 = 25 = b^2","$ 5^2 = 25 = b^2 $","inlinerMath-tDEurJ8Oh",{"__TSPROSE_proseElement":213,"schema":2453,"data":1867},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-gcBVGuLlE",{"__TSPROSE_proseElement":213,"schema":2456,"data":2457,"storageKey":2368,"id":2458},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2367,"freeze":217},"blockMath-xBGmZf40t-1","primery-ispolzovaniya-kvadrata-raznosti","square-sum-diff-details",{"__TSPROSE_proseElement":213,"schema":2462},{"name":2463,"type":216,"linkable":217,"__TSPROSE_schema":213},"hr",{"__TSPROSE_proseElement":213,"schema":2465,"data":2467,"children":2470,"id":2482},{"name":2466,"type":216,"linkable":213,"__TSPROSE_schema":213},"accent_term",{"title":2468,"layout":2469},"Формулы сокращённого умножения","column",[2471],{"__TSPROSE_proseElement":213,"schema":2472,"children":2474},{"name":2473,"type":216,"linkable":217,"__TSPROSE_schema":213},"accentMain_term",[2475],{"__TSPROSE_proseElement":213,"schema":2476,"children":2477,"id":2481},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2478],{"__TSPROSE_proseElement":213,"schema":2479,"data":2480},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Формулы, которые позволяют быстро \"разворачивать\" компактные выражения со степенями в какое-то разложение или наоборот, \"сворачивать\" длинные суммы в компактную форму. Эти формулы нужны, чтобы не тратить время на рутинные вычисления вручную.","paragraph-sGVpa2xZy","formuly-sokraschennogo-umnozheniya",{"__TSPROSE_proseElement":213,"schema":2484},{"name":2463,"type":216,"linkable":217,"__TSPROSE_schema":213},{"__TSPROSE_proseElement":213,"schema":2486,"data":2488,"children":2490,"id":2698},{"name":2487,"type":216,"linkable":213,"__TSPROSE_schema":213},"list",{"type":2489},"unordered",[2491,2565,2626],{"__TSPROSE_proseElement":213,"schema":2492,"children":2494},{"name":2493,"type":216,"linkable":217,"__TSPROSE_schema":213},"listItem",[2495,2507,2530,2536],{"__TSPROSE_proseElement":213,"schema":2496,"children":2497,"id":2506},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2498],{"__TSPROSE_proseElement":213,"schema":2499,"data":2500,"children":2501,"id":2505},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[2502],{"__TSPROSE_proseElement":213,"schema":2503,"data":2504},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Знак плюс-минус","emphasis-1T20LvUmV","paragraph-1JIYzh2eP",{"__TSPROSE_proseElement":213,"schema":2508,"children":2509,"id":2529},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2510,2513,2519,2522,2526],{"__TSPROSE_proseElement":213,"schema":2511,"data":2512},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Не запоминайте по отдельности ",{"__TSPROSE_proseElement":213,"schema":2514,"data":2515,"storageKey":2517,"id":2518},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2516},"4","$ 4 $","inlinerMath-ZXyaHjumL",{"__TSPROSE_proseElement":213,"schema":2520,"data":2521},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," формулы: квадрат суммы, квадрат разности, куб суммы и куб разности. Достаточно запомнить ",{"__TSPROSE_proseElement":213,"schema":2523,"data":2524,"storageKey":543,"id":2525},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":542},"inlinerMath-z9Yar9waf-5",{"__TSPROSE_proseElement":213,"schema":2527,"data":2528},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," формулы, если использовать знак плюс-минус, ведь кроме знаков больше ничего в них не меняется:","paragraph-TP6k3DSmh",{"__TSPROSE_proseElement":213,"schema":2531,"data":2532,"storageKey":2534,"id":2535},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2533,"freeze":217},"(a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3","$$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $$","blockMath-kiNVg4zvS",{"__TSPROSE_proseElement":213,"schema":2537,"children":2538,"id":2564},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2539,2542,2550,2553,2561],{"__TSPROSE_proseElement":213,"schema":2540,"data":2541},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"В суммах все знаки ",{"__TSPROSE_proseElement":213,"schema":2543,"data":2544,"children":2545,"id":2549},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[2546],{"__TSPROSE_proseElement":213,"schema":2547,"data":2548},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"всегда плюсы","emphasis-WATBgUbHA",{"__TSPROSE_proseElement":213,"schema":2551,"data":2552},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},". В разности ",{"__TSPROSE_proseElement":213,"schema":2554,"data":2555,"children":2556,"id":2560},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[2557],{"__TSPROSE_proseElement":213,"schema":2558,"data":2559},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"минус всегда идет сразу","emphasis-DzWKDXE2b",{"__TSPROSE_proseElement":213,"schema":2562,"data":2563},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," после первого слагаемого в разложении. В случае куба еще и у последнего слагаемого.","paragraph-NJkC7n6aH",{"__TSPROSE_proseElement":213,"schema":2566,"children":2567},{"name":2493,"type":216,"linkable":217,"__TSPROSE_schema":213},[2568,2580,2593],{"__TSPROSE_proseElement":213,"schema":2569,"children":2570,"id":2579},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2571],{"__TSPROSE_proseElement":213,"schema":2572,"data":2573,"children":2574,"id":2578},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[2575],{"__TSPROSE_proseElement":213,"schema":2576,"data":2577},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Разность квадратов особняком","emphasis-ealRCqFXE","paragraph-vFqUVvk41",{"__TSPROSE_proseElement":213,"schema":2581,"children":2582,"id":2592},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2583,2586],{"__TSPROSE_proseElement":213,"schema":2584,"data":2585},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Формулы, названия которых начинаются со степени (\"квадрат ...\" и \"куб ...\"), имеют похожую форму и их можно вывести естественным образом через раскрытие скобок ",{"__TSPROSE_proseElement":213,"schema":2587,"data":2588,"storageKey":2590,"id":2591},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2589},"(a+b)^2 = (a+b)(a+b) = \\ldots","$ (a+b)^2 = (a+b)(a+b) = \\ldots $","inlinerMath-E2hB8EXr8","paragraph-GM7HAKhf4",{"__TSPROSE_proseElement":213,"schema":2594,"children":2595,"id":2625},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2596,2599,2605,2608,2614,2617,2623],{"__TSPROSE_proseElement":213,"schema":2597,"data":2598},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"А вот разность квадратов ",{"__TSPROSE_proseElement":213,"schema":2600,"data":2601,"storageKey":2603,"id":2604},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2602},"a^2-b^2","$ a^2-b^2 $","inlinerMath-Z18OlR0cJ",{"__TSPROSE_proseElement":213,"schema":2606,"data":2607},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," стоит особняком. Во-первых, она раскладывается в скобки с плюсом и минусом. Во-вторых из формы ",{"__TSPROSE_proseElement":213,"schema":2609,"data":2610,"storageKey":2612,"id":2613},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2611},"a^2 - b^2","$ a^2 - b^2 $","inlinerMath-T1YOXnC6X",{"__TSPROSE_proseElement":213,"schema":2615,"data":2616},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," нельзя естественным и явным образом получить из самой разности произведение ",{"__TSPROSE_proseElement":213,"schema":2618,"data":2619,"storageKey":2621,"id":2622},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2620},"(a+b)(a-b)","$ (a+b)(a-b) $","inlinerMath-bZ4wF1qyF",{"__TSPROSE_proseElement":213,"schema":2624,"data":975},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-TjiEV1CkX",{"__TSPROSE_proseElement":213,"schema":2627,"children":2628},{"name":2493,"type":216,"linkable":217,"__TSPROSE_schema":213},[2629,2641,2663,2669],{"__TSPROSE_proseElement":213,"schema":2630,"children":2631,"id":2640},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2632],{"__TSPROSE_proseElement":213,"schema":2633,"data":2634,"children":2635,"id":2639},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[2636],{"__TSPROSE_proseElement":213,"schema":2637,"data":2638},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"Какая степень -- такой и коэффициент","emphasis-KSb50gTxe","paragraph-DJoU9TSeB",{"__TSPROSE_proseElement":213,"schema":2642,"children":2643,"id":2662},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2644,2647,2651,2654,2660],{"__TSPROSE_proseElement":213,"schema":2645,"data":2646},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"В квадрате\u002Fкубе суммы\u002Fразности степень (вторая или третья) встречается и как коэффициент в разложении. Для квадрата суммы\u002Fразности это ",{"__TSPROSE_proseElement":213,"schema":2648,"data":2649,"storageKey":543,"id":2650},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":542},"inlinerMath-z9Yar9waf-6",{"__TSPROSE_proseElement":213,"schema":2652,"data":2653},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},", для куба суммы\u002Fразности это ",{"__TSPROSE_proseElement":213,"schema":2655,"data":2656,"storageKey":2658,"id":2659},{"name":280,"type":226,"linkable":213,"__TSPROSE_schema":213},{"katex":2657},"3","$ 3 $","inlinerMath-mqn34wvB0",{"__TSPROSE_proseElement":213,"schema":2661,"data":777},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"paragraph-HH041xis9",{"__TSPROSE_proseElement":213,"schema":2664,"data":2665,"storageKey":2667,"id":2668},{"name":409,"type":216,"linkable":213,"__TSPROSE_schema":213},{"katex":2666,"freeze":217},"(a \\pm b)^{\\normalsize\\brand{2}} = a^2 \\pm \\brand{2}ab + b^2 \\\\ (a \\pm b)^{\\normalsize\\brand{3}} = a^3 \\pm \\brand{3}a^2b + \\brand{3}ab^2 \\pm b^3","$$ (a \\pm b)^{\\normalsize\\brand{2}} = a^2 \\pm \\brand{2}ab + b^2 \\\\ (a \\pm b)^{\\normalsize\\brand{3}} = a^3 \\pm \\brand{3}a^2b + \\brand{3}ab^2 \\pm b^3 $$","blockMath-L9UAp9IUH",{"__TSPROSE_proseElement":213,"schema":2670,"children":2671,"id":2697},{"name":392,"type":216,"linkable":213,"__TSPROSE_schema":213},[2672,2675,2683,2686,2694],{"__TSPROSE_proseElement":213,"schema":2673,"data":2674},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"А еще коэффициент легко вспоминть по геометрическому выводу формул. Для \"квадрата ...\" мы составляем квадрат и в процессе появляются ",{"__TSPROSE_proseElement":213,"schema":2676,"data":2677,"children":2678,"id":2682},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[2679],{"__TSPROSE_proseElement":213,"schema":2680,"data":2681},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"два","emphasis-lF0cBJQwp",{"__TSPROSE_proseElement":213,"schema":2684,"data":2685},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," прямоугольника. А для \"куба ...\" мы составляем куб и в процессе появляются два вида из ",{"__TSPROSE_proseElement":213,"schema":2687,"data":2688,"children":2689,"id":2693},{"name":236,"type":226,"linkable":213,"__TSPROSE_schema":213},{"type":238},[2690],{"__TSPROSE_proseElement":213,"schema":2691,"data":2692},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213},"трех","emphasis-dhAIDBC6v",{"__TSPROSE_proseElement":213,"schema":2695,"data":2696},{"name":242,"type":226,"linkable":217,"__TSPROSE_schema":213}," параллелепипедов.","paragraph-rze0H5eSn","kak-zapomnit-fsu",{"$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 $":2700,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fsummary\u002F$squareSumDiffDetails":2701,"$ a^2 - b^2 = (a+b)(a-b) $":2700,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$diffOfSquaresExamples":2707,"$ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $":2700,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$cubeSumDiffExamples":2713,"$$ (m+5)^2 $$":2700,"$$ m^2 + 10m + 25 $$":2700,"$ a $":2700,"$ m $":2700,"$ b $":2700,"$ 5 $":2700,"$$ m^2 >>{big} 2 \\cdot m \\cdot 5 = 10m >>{big} 5^2 = 25 $$":2700,"$$ (m+5)^2 = m^2 + 10m + 25 $$":2700,"$$ \\left( 2 + \\frac{1}{8}x \\right)^2 $$":2700,"$$ 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$":2700,"$ 2 $":2700,"$ \\frac{1}{8}x $":2700,"$$ 2^2 = 4 >>{big} \\cancel{2} \\cdot \\cancel{2} \\cdot \\frac{1}{\\cancel{8}_{\\small\\cancel{4}_{\\small 2}}}x = \\frac{1}{2}x >>{big} \\left( \\frac{1}{8}x \\right)^2 = \\left(\\frac{1}{8}\\right)^2 x^2 = \\frac{1}{64}x^2 $$":2700,"$$ \\left( 2 + \\frac{1}{8}x \\right)^2 = 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$":2700,"$$ (-7a - 3b)^2 $$":2700,"$ 1 $":2700,"$$ (-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2 $$":2700,"$$ 49a^2 + 42ab + 9b^2 $$":2700,"$ -7a - 3b $":2700,"$ -7a $":2700,"$ 3b $":2700,"$ -3b $":2700,"$$ (-7a)^2 = (-7)^2 a^2 = 49a^2 >>{big} 2 \\cdot (-7a) \\cdot (-3b) = 42ab >>{big} (-3b)^2 = (-3)^2 b^2 = 9b^2 $$":2700,"$$ (-7a - 3b)^2 = 49a^2 + 42ab + 9b^2 $$":2700,"$$ -7a - 3b = -(7a + 3b) $$":2700,"$$ (-(7a + 3b))^2 = (-1)^2 \\cdot (7a + 3b)^2 = (7a + 3b)^2 = 49a^2 + 42ab + 9b^2 $$":2700,"$$ 49 + 14x + x^2 $$":2700,"$$ (7 + x)^2 $$":2700,"$ 49 = 7^2 $":2700,"$ x^2 = x^2 $":2700,"$ a = 7 $":2700,"$ b = x $":2700,"$ 2ab = 2 \\cdot 7 \\cdot x = 14x $":2700,"$$ 49 + 14x + x^2 = \\underset{a^2}{7^2} + 2 \\cdot \\underset{a}{7} \\cdot \\underset{b}{x} + \\underset{b^2}{x^2} = (7 + x)^2 $$":2700,"$ 14x $":2700,"$ 7x $":2700,"$ 49 $":2700,"$ x^2 $":2700,"$ 7x = 7 \\cdot x $":2700,"$$ 1 + 8y + 16y^2 $$":2700,"$ 1^2 = 1 $":2700,"$$ (1 + 4y)^2 $$":2700,"$ 1 = 1^2 $":2700,"$ 16y^2 = (4y)^2 $":2700,"$ a = 1 $":2700,"$ b = 4y $":2700,"$ 2ab = 2 \\cdot 1 \\cdot 4y = 8y $":2700,"$$ 1 + 8y + 16y^2 = \\underset{a^2}{1^2} + 2 \\cdot \\underset{a}{1} \\cdot \\underset{b}{4y} + \\underset{b^2}{(4y)^2} = (1 + 4y)^2 $$":2700,"$ 8y $":2700,"$ 4y $":2700,"$ 16y^2 $":2700,"$ 1 \\cdot 4y $":2700,"$ 1^2 = 1 = a $":2700,"$ (4y)^2 = 16y^2 = b^2 $":2700,"$$ \\frac{1}{4}k^2 + k + 1 $$":2700,"$$ \\left(\\frac{k}{2} + 1 \\right)^2 $$":2700,"$ \\frac{1}{4}k^2 = \\left(\\frac{k}{2}\\right)^2 $":2700,"$ a = \\frac{k}{2} $":2700,"$ b = 1 $":2700,"$ 2ab = 2 \\cdot \\frac{k}{2} \\cdot 1 = k $":2700,"$$ \\frac{1}{4}k^2 + k + 1 = \\underset{a^2}{\\left( \\frac{k}{2} \\right)^2} + 2 \\cdot \\underset{a}{\\frac{k}{2}} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = \\left(\\frac{k}{2} + 1 \\right)^2 $$":2700,"$ k $":2700,"$ \\frac{k}{2} $":2700,"$ \\frac{1}{4}k^2 $":2700,"$ \\frac{k}{2} \\cdot 1 $":2700,"$ \\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2 $":2700,"$ 1^2 = 1 = b^2 $":2700,"$$ 16t^2 + 36m^2 + 48tm $$":2700,"$$ (4t + 6m)^2 $$":2700,"$$ 16t^2 + 48tm + 36m^2 $$":2700,"$ 16t^2 = (4t)^2 $":2700,"$ 36m^2 = (6m)^2 $":2700,"$ a = 4t $":2700,"$ b = 6m $":2700,"$ 2ab = 2 \\cdot 4t \\cdot 6m = 48tm $":2700,"$$ 16t^2 + 36m^2 + 48tm = \\underset{a^2}{(4t)^2} + 2 \\cdot \\underset{a}{4t} \\cdot \\underset{b}{6m} + \\underset{b^2}{(6m)^2} = (4t + 6m)^2 $$":2700,"$ 48tm $":2700,"$ 24tm $":2700,"$ 16t^2 $":2700,"$ 36m^2 $":2700,"$ 24tm = 4t \\cdot 6m $":2700,"$ (4t)^2 = 16t^2 = a^2 $":2700,"$ (6m)^2 = 36m^2 = b^2 $":2700,"$$ (6-c)^2 $$":2700,"$$ 36 - 12c + c^2 $$":2700,"$ 6 $":2700,"$ c $":2700,"$$ 6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2 $$":2700,"$$ (6-c)^2 = 36 - 12c + c^2 $$":2700,"$ -b $":2700,"$$ (6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2} $$":2700,"$$ 9x^2 - 6x + 1 $$":2700,"$$ (3x - 1)^2 $$":2700,"$ 9x^2 = (3x)^2 $":2700,"$ a = 3x $":2700,"$ -2ab = -2 \\cdot 3x \\cdot 1 = -6x $":2700,"$$ 9x^2 - 6x + 1 = \\underset{a^2}{(3x)^2} - 2 \\cdot \\underset{a}{3x} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = (3x - 1)^2 $$":2700,"$ -6x $":2700,"$ -2 $":2700,"$ 3x $":2700,"$ 9x^2 $":2700,"$ 3x = 3x \\cdot 1 $":2700,"$ 9x^2 = (3x)^2 = a^2 $":2700,"$ 1 = 1^2 = b^2 $":2700,"$$ (-7 + 2a)^2 $$":2700,"$$ 4a^2 - 28a + 49 $$":2700,"$$ (-7 + 2a)^2 = (2a - 7)^2 $$":2700,"$ 2a $":2700,"$ 7 $":2700,"$$ (2a)^2 = 4a^2 >>{big} -2 \\cdot 2a \\cdot 7 = -28a >>{big} 7^2 = 49 $$":2700,"$$ (-7 + 2a)^2 = 4a^2 - 28a + 49 $$":2700,"$$ - 12k + 4k^2 + 9 $$":2700,"$$ (2k - 3)^2 $$":2700,"$$ - 12k + 4k^2 + 9 = 4k^2 - 12k + 9 $$":2700,"$ 4k^2 = (2k)^2 $":2700,"$ 9 = 3^2 $":2700,"$ a = 2k $":2700,"$ b = 3 $":2700,"$ -2ab = -2 \\cdot 2k \\cdot 3 = -12k $":2700,"$$ 4k^2 - 12k + 9 = \\underset{a^2}{(2k)^2} - 2 \\cdot \\underset{a}{2k} \\cdot \\underset{b}{3} + \\underset{b^2}{3^2} = (2k - 3)^2 $$":2700,"$ -12k $":2700,"$ 6k $":2700,"$ 4k^2 $":2700,"$ 9 $":2700,"$ 6k = 2k \\cdot 3 $":2700,"$ (2k)^2 = 4k^2 = a^2 $":2700,"$ 3^2 = 9 = b^2 $":2700,"$$ \\left( 5y - \\frac{3}{4}x \\right)^2 $$":2700,"$$ 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$":2700,"$ 5y $":2700,"$ \\frac{3}{4}x $":2700,"$$ (5y)^2 = 5^2y^2 = 25y^2 >>{big} -\\cancel{2} \\cdot 5y \\cdot \\frac{3}{\\cancel{4}_{\\small 2}}x = -\\frac{15}{2}xy >>{big} \\left( \\frac{3}{4}x \\right)^2 = \\left(\\frac{3}{4}\\right)^2 x^2 = \\frac{9}{16}x^2 $$":2700,"$$ \\left( 5y - \\frac{3}{4}x \\right)^2 = 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$":2700,"$$ \\frac{1}{4}m^2 - 5m + 25 $$":2700,"$$ \\left(\\frac{m}{2} - 5\\right)^2 $$":2700,"$ \\frac{1}{4}m^2 = \\left(\\frac{m}{2}\\right)^2 $":2700,"$ 25 = 5^2 $":2700,"$ a = \\frac{m}{2} $":2700,"$ b = 5 $":2700,"$ -2ab = -2 \\cdot \\frac{m}{2} \\cdot 5 = -5m $":2700,"$$ \\frac{1}{4}m^2 - 5m + 25 = \\underset{a^2}{\\left(\\frac{1}{2}m\\right)^2} - 2 \\cdot \\underset{a}{\\frac{1}{2}m} \\cdot \\underset{b}{5} + \\underset{b^2}{5^2} = \\left(\\frac{m}{2} - 5\\right)^2 $$":2700,"$ -5m $":2700,"$ \\frac{5m}{2} $":2700,"$ \\frac{1}{4}m^2 $":2700,"$ 25 $":2700,"$ \\frac{5m}{2} = \\frac{m}{2} \\cdot 5 $":2700,"$ \\left(\\frac{m}{2}\\right)^2 = \\frac{1}{4}m^2 = a^2 $":2700,"$ 5^2 = 25 = b^2 $":2700,"$ 4 $":2700,"$$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $$":2700,"$ (a+b)^2 = (a+b)(a+b) = \\ldots $":2700,"$ a^2-b^2 $":2700,"$ a^2 - b^2 $":2700,"$ (a+b)(a-b) $":2700,"$ 3 $":2700,"$$ (a \\pm b)^{\\normalsize\\brand{2}} = a^2 \\pm \\brand{2}ab + b^2 \\\\ (a \\pm b)^{\\normalsize\\brand{3}} = a^3 \\pm \\brand{3}a^2b + \\brand{3}ab^2 \\pm b^3 $$":2700},null,{"type":2702,"schemaName":377,"elementTitle":380,"content":2703,"resolvedHref":2704,"previewRequest":2705},"unique",{"contentType":174,"contentTitle":153,"topicPart":208},"\u002Fsummary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=square-sum-diff-details",{"type":2702,"contentFullId":151,"contentType":174,"topicPart":208,"uniqueName":2706},"squareSumDiffDetails",{"type":2702,"schemaName":384,"elementTitle":2708,"content":2709,"resolvedHref":2710,"previewRequest":2711},"Примеры использования разности квадратов",{"contentType":174,"contentTitle":153,"topicPart":210},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=diff-of-squares-examples",{"type":2702,"contentFullId":151,"contentType":174,"topicPart":210,"uniqueName":2712},"diffOfSquaresExamples",{"type":2702,"schemaName":384,"elementTitle":2714,"content":2715,"resolvedHref":2716,"previewRequest":2717},"Примеры использования кубов суммы и разности",{"contentType":174,"contentTitle":153,"topicPart":210},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum-diff-examples",{"type":2702,"contentFullId":151,"contentType":174,"topicPart":210,"uniqueName":2718},"cubeSumDiffExamples",[2720,2727,2732,2737,2742,2748,2755,2760,2766,2772,2779,2786,2791,2796],{"link":2721,"schemaName":2722,"title":2723,"seo":2724,"description":2726},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=chto-takoe-fsu","heading","Что такое ФСУ?",{"title":2725},"Что такое формулы сокращенного умножения (ФСУ)?","Формулы сокращенного умножения (ФСУ) - это набор формул, которые позволяют быстро раскрывать скобки или же наоборот, запаковывать выражения в скобки.",{"link":2728,"schemaName":2466,"title":2468,"key":2729,"seo":2730,"description":2731},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=what-are-special-products",{"title":2723},{},"Формулы, которые позволяют быстро \"разворачивать\" компактные выражения со степенями в какое-то разложение или\n          наоборот, \"сворачивать\" длинные суммы в компактную форму. Эти формулы нужны, чтобы не тратить время на рутинные\n          вычисления вручную.",{"link":181,"schemaName":182,"title":2733,"key":2734,"seo":2735,"description":2736},"Квадрат суммы",{},{},"Одна из формул сокращенного умножения (ФСУ): (a+b)² = a² + 2ab + b².\n          Позволяет быстро раскрывать скобки или же наоборот, запаковывать в них уже разложенные выражения.",{"link":186,"schemaName":182,"title":2738,"key":2739,"seo":2740,"description":2741},"Квадрат разности",{},{},"Одна из формул сокращенного умножения (ФСУ): (a-b)² = a² - 2ab + b².\n          Позволяет быстро раскрывать скобки или же наоборот, запаковывать в них уже разложенные выражения.",{"link":2743,"schemaName":2722,"title":2744,"seo":2745,"description":2747},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=primenenie-kvadrata-summy-ili-raznosti","Применение квадрата суммы или разности",{"title":2746},"Где в жизни используются квадрат суммы и разности?","Практические примеры применения формул сокращенного умножения (ФСУ) для квадрата суммы и разности в различных областях математики и реальной жизни.",{"link":2749,"schemaName":384,"title":2750,"key":2751,"seo":2752,"description":2754},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=fast-square","Быстрые квадраты",{},{"title":2753},"Как быстро возвести число в квадрат в уме?","Универсальный метод быстрого возведения в квадрат любого числа с помощью формул квадрата суммы и разности.\n          Для небольших чисел в пределах 100 можно выполнять этот процесс даже в уме!",{"link":2756,"schemaName":182,"title":307,"key":2757,"seo":2758,"description":2759},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=diff-of-squares",{},{},"Одна из формул сокращенного умножения (ФСУ): a² - b² = (a + b)(a - b).\n          Позволяет вдвое снизить или увеличить степень.\n          Используется для упрощения выражений.",{"link":2761,"schemaName":182,"title":2762,"key":2763,"seo":2764,"description":2765},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum","Куб суммы",{},{},"Одна из формул сокращенного умножения (ФСУ): (a + b)³ = a³ + 3a²b + 3ab² + b³.\n          Позволяет быстро разложить куб суммы двух выражений.\n          Используется для упрощения выражений и решения уравнений.",{"link":2767,"schemaName":182,"title":2768,"key":2769,"seo":2770,"description":2771},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-diff","Куб разности",{},{},"Одна из формул сокращенного умножения (ФСУ): (a - b)³ = a³ - 3a²b + 3ab² - b³.\n          Позволяет быстро разложить куб разности двух выражений.\n          Используется для упрощения выражений и решения уравнений.",{"link":2773,"schemaName":2487,"title":2774,"key":2775,"seo":2776,"description":2778},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=memorization-tips","Как запомнить ФСУ?",{},{"title":2777},"Как запомнить формулы сокращенного умножения?","Набор советов и приемов, которые помогут наизусть запомнить формулы сокращенного умножения (ФСУ) и не путаться в них.",{"link":2780,"schemaName":221,"title":2781,"key":2782,"seo":2784,"description":2785},"\u002Fsummary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=special-products-table","Таблица формул сокращенного умножения",{"title":2783},"Таблица ФСУ",{},"Единая таблица со всеми формулами сокращенного умножения.",{"link":2787,"schemaName":384,"title":2788,"seo":2789,"description":2790},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=otrabotka-kvadrata-summy-i-raznosti","Задачи на квадрат суммы и разности",{},"Большое количество задач на отработку формул сокращенного умножения -- квадрата суммы и разности.\n          Два типа задач -- раскрыть скобки и запаковать в квадрат суммы или разности.",{"link":2792,"schemaName":384,"title":2793,"seo":2794,"description":2795},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=otrabotka-raznosti-kvadratov","Задачи на разность квадратов",{},"Большое количество задач на отработку формулы разности квадратов.\n          Два типа задач -- раскрыть скобки и запаковать в разность квадратов.",{"link":2797,"schemaName":384,"title":2798,"seo":2799,"description":2800},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=otrabotka-kuba-summy-i-raznosti","Задачи на куб суммы и разности",{},"Большое количество задач на отработку формул сокращенного умножения -- куба суммы и разности.\n          Два типа задач -- раскрыть скобки и запаковать в куб суммы или разности.",[2802,2804,2805],{"type":2803,"schemaName":221,"title":2783,"elementId":374},"element",{"type":2803,"schemaName":2466,"title":2723,"elementId":2482},{"type":2803,"schemaName":2487,"title":2774,"elementId":2698},{"contributorsCount":2807,"sponsorsCount":167},7,{"shortBookId":5,"frontNav":2809},{"type":14,"shortId":5,"title":15,"flags":2810,"link":17,"children":2811},{},[2812,2821],{"type":2813,"separator":213,"shortId":2814,"title":159,"flags":2815,"link":160,"children":2816},"group","foundations\u002Fpolynomials",{},[2817],{"type":174,"shortId":151,"title":153,"flags":2818,"link":2819,"parts":2820},{},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F",[210,208,211],{"type":2813,"separator":213,"shortId":2822,"title":2823,"flags":2824,"link":2825,"children":2826},"foundations\u002Fequations","Уравнения",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002F",[2827,2833,2838],{"type":174,"shortId":2828,"title":2829,"flags":2830,"link":2831,"parts":2832},"foundations\u002Fequations\u002Felementary","Элементарные уравнения",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Felementary\u002F",[210,208,211],{"type":8,"shortId":2834,"title":2835,"flags":2836,"link":2837},"foundations\u002Fequations\u002Fzero-product-property","Нулевые множители",{},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fzero-product-property\u002F",{"type":2813,"separator":217,"shortId":2839,"title":2840,"flags":2841,"link":2842,"children":2843},"foundations\u002Fequations\u002Fquadratic","Квадратные уравнения",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002Fquadratic\u002F",[2844,2850,2856,2861,2867,2873,2878,2884,2890],{"type":174,"shortId":2845,"title":2846,"flags":2847,"link":2848,"parts":2849},"foundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it","Что это такое?",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it\u002F",[210,208,211],{"type":174,"shortId":2851,"title":2852,"flags":2853,"link":2854,"parts":2855},"foundations\u002Fequations\u002Fquadratic\u002Fincomplete","Неполная форма",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fincomplete\u002F",[210,208,211],{"type":174,"shortId":2857,"title":2858,"flags":2859,"link":176,"parts":2860},"foundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square","Полный квадрат",{},[210,208,211],{"type":174,"shortId":2862,"title":2863,"flags":2864,"link":2865,"parts":2866},"foundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula","Формула корней",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002F",[210,208,211],{"type":174,"shortId":2868,"title":2869,"flags":2870,"link":2871,"parts":2872},"foundations\u002Fequations\u002Fquadratic\u002Ffactoring","Разложение на множители",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002F",[210,208,211],{"type":8,"shortId":2874,"title":2875,"flags":2876,"link":2877},"foundations\u002Fequations\u002Fquadratic\u002Freal-world","В реальной жизни",{"secondary":213},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F",{"type":174,"shortId":2879,"title":2880,"flags":2881,"link":2882,"parts":2883},"foundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas","Теорема Виета",{"secondary":213},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas\u002F",[210,208,211],{"type":174,"shortId":2885,"title":2886,"flags":2887,"link":2888,"parts":2889},"foundations\u002Fequations\u002Fquadratic\u002Fmental-solving","Решение в уме",{"secondary":213,"advanced":213},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving\u002F",[210,208],{"type":174,"shortId":2891,"title":2892,"flags":2893,"link":2894,"parts":2895},"foundations\u002Fequations\u002Fquadratic\u002Fformulas","Общие формулы",{"secondary":213},"\u002Fsummary\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fformulas\u002F",[208,211],{"cameo":2897,"sponsor":2908},[2898,2899,2900,2901,2902,2903,2904,2905,2906,2907],"cameo-black-hole","cameo-confucius","cameo-dunno","cameo-einstein","cameo-euclid","cameo-gagarin","cameo-gauss","cameo-gorinich","cameo-newton","cameo-t800",[2909,2910],"sponsor-green-hedgehog","sponsor-reverse-shift",1777651272391]