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b)(a-b) = a^2 - ab - ba + b^2 = a^2 - 2ab + b^2","$$ (a+b)^2 = (a + b)(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2 \\\\ (a-b)^2 = (a - b)(a-b) = a^2 - ab - ba + b^2 = a^2 - 2ab + b^2 $$","blockMath-jnqu9zQsz",{"__TSPROSE_proseElement":219,"schema":415,"children":416,"id":420},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[417],{"__TSPROSE_proseElement":219,"schema":418,"data":419},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Вычтем второе из первого:","paragraph-q6L92Wp7E",{"__TSPROSE_proseElement":219,"schema":422,"data":423,"storageKey":425,"id":426},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":424,"freeze":223},"(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = \\cancel{a^2} + 2ab + \\cancel{b^2} - \\cancel{a^2} + 2ab - \\cancel{b^2} = 4ab","$$ (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = \\cancel{a^2} + 2ab + \\cancel{b^2} - \\cancel{a^2} + 2ab - \\cancel{b^2} = 4ab $$","blockMath-O0p7EERnc",{"__TSPROSE_proseElement":219,"schema":428,"children":429,"id":442},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[430,433,439],{"__TSPROSE_proseElement":219,"schema":431,"data":432},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Числитель целиком свернулся в ",{"__TSPROSE_proseElement":219,"schema":434,"data":435,"storageKey":437,"id":438},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":436},"4ab","$ 4ab $","inlinerMath-qcb9CkvtK",{"__TSPROSE_proseElement":219,"schema":440,"data":441},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," -- ровно то, что стоит в знаменателе:","paragraph-mcxJEcNHm",{"__TSPROSE_proseElement":219,"schema":444,"data":445,"storageKey":447,"id":448},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":446,"freeze":223},"\\frac{\\cancel{4ab}}{\\cancel{4ab}} = \\boxed{1}","$$ \\frac{\\cancel{4ab}}{\\cancel{4ab}} = \\boxed{1} $$","blockMath-bZWeRNYuh",{"__TSPROSE_proseElement":219,"schema":450,"data":451,"children":452},{"name":258,"type":222,"linkable":223,"__TSPROSE_schema":219},{},[453,462,468,476],{"__TSPROSE_proseElement":219,"schema":454,"children":455},{"name":263,"type":222,"linkable":223,"__TSPROSE_schema":219},[456],{"__TSPROSE_proseElement":219,"schema":457,"data":458,"storageKey":460,"id":461},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":459,"freeze":223},"\\frac{(m+n)^2 - m^2 - n^2}{mn} \\cdot \\frac{(m-n)^2 - m^2 - n^2}{mn}","$$ \\frac{(m+n)^2 - m^2 - n^2}{mn} \\cdot \\frac{(m-n)^2 - m^2 - n^2}{mn} $$","blockMath-xj0IdLwAL",{"__TSPROSE_proseElement":219,"schema":463,"data":464},{"name":274,"type":222,"linkable":223,"__TSPROSE_schema":219},{"serializedValidator":465},{"__ERUDIT_CHECK":219,"name":277,"data":466},{"expr":467},"-4",{"__TSPROSE_proseElement":219,"schema":469,"children":470},{"name":282,"type":222,"linkable":223,"__TSPROSE_schema":219},[471],{"__TSPROSE_proseElement":219,"schema":472,"data":473,"storageKey":474,"id":475},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":467,"freeze":223},"$$ -4 $$","blockMath-0OjzEYrdG",{"__TSPROSE_proseElement":219,"schema":477,"children":478},{"name":291,"type":222,"linkable":223,"__TSPROSE_schema":219},[479,486,492,499,505,512],{"__TSPROSE_proseElement":219,"schema":480,"children":481,"id":485},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[482],{"__TSPROSE_proseElement":219,"schema":483,"data":484},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Разберёмся с каждым числителем отдельно. Раскроем квадрат в первом:","paragraph-HdAIPBm85",{"__TSPROSE_proseElement":219,"schema":487,"data":488,"storageKey":490,"id":491},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":489,"freeze":223},"(m+n)^2 - m^2 - n^2 = \\cancel{m^2} + 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = 2mn","$$ (m+n)^2 - m^2 - n^2 = \\cancel{m^2} + 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = 2mn $$","blockMath-tPacNDWyD",{"__TSPROSE_proseElement":219,"schema":493,"children":494,"id":498},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[495],{"__TSPROSE_proseElement":219,"schema":496,"data":497},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"И во втором:","paragraph-EY1Z60lds",{"__TSPROSE_proseElement":219,"schema":500,"data":501,"storageKey":503,"id":504},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":502,"freeze":223},"(m-n)^2 - m^2 - n^2 = \\cancel{m^2} - 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = -2mn","$$ (m-n)^2 - m^2 - n^2 = \\cancel{m^2} - 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = -2mn $$","blockMath-4Ap7uaTLg",{"__TSPROSE_proseElement":219,"schema":506,"children":507,"id":511},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[508],{"__TSPROSE_proseElement":219,"schema":509,"data":510},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Подставляем обратно и сокращаем:","paragraph-MGtZMKfHM",{"__TSPROSE_proseElement":219,"schema":513,"data":514,"storageKey":516,"id":517},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":515,"freeze":223},"\\frac{2\\cancel{mn}}{\\cancel{mn}} \\cdot \\frac{-2\\cancel{mn}}{\\cancel{mn}} = 2 \\cdot (-2) = \\boxed{-4}","$$ \\frac{2\\cancel{mn}}{\\cancel{mn}} \\cdot \\frac{-2\\cancel{mn}}{\\cancel{mn}} = 2 \\cdot (-2) = \\boxed{-4} $$","blockMath-9vwuJBOgx","intro-examples",{"__TSPROSE_proseElement":219,"schema":520,"children":521,"id":534},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[522,525,531],{"__TSPROSE_proseElement":219,"schema":523,"data":524},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Во всех трёх примерах мы раз за разом проводили рутинные действия -- раскрытие скобок в каких-то степенях. Результаты раскрытия скобок похоже друг на друга -- квадраты, коэффициенты ",{"__TSPROSE_proseElement":219,"schema":526,"data":527,"storageKey":529,"id":530},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":528},"2","$ 2 $","inlinerMath-z9Yar9waf",{"__TSPROSE_proseElement":219,"schema":532,"data":533},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," и так далее. Чтобы каждый раз не тратить время на рутинные вычисления вручную умные математики самые часто встречающиеся ситуации разложения изучили и оформили в виде формул. Отсюда и название -- формулы сокращенного умножения (ФСУ). Не путать с ФСО и ФСБ!","paragraph-qcjP2crF6",{"__TSPROSE_proseElement":219,"schema":536,"data":538,"children":541,"id":553},{"name":537,"type":222,"linkable":219,"__TSPROSE_schema":219},"accent_term",{"title":539,"layout":540},"Формулы сокращённого умножения","column",[542],{"__TSPROSE_proseElement":219,"schema":543,"children":545},{"name":544,"type":222,"linkable":223,"__TSPROSE_schema":219},"accentMain_term",[546],{"__TSPROSE_proseElement":219,"schema":547,"children":548,"id":552},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[549],{"__TSPROSE_proseElement":219,"schema":550,"data":551},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"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","what-are-special-products",{"__TSPROSE_proseElement":219,"schema":555,"data":556,"id":558},{"name":227,"type":222,"linkable":219,"__TSPROSE_schema":219},{"level":172,"title":557},"Квадрат суммы","kvadrat-summy",{"__TSPROSE_proseElement":219,"schema":560,"children":561,"id":574},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[562,565,571],{"__TSPROSE_proseElement":219,"schema":563,"data":564},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Начнём с классической ошибки, которую допускают ",{"__TSPROSE_proseElement":219,"schema":566,"data":567,"storageKey":569,"id":570},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":568},"90%","$ 90% $","inlinerMath-TriCVV86T",{"__TSPROSE_proseElement":219,"schema":572,"data":573},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," людей, чаще всего школьников, у которых \"не очень\" с математикой. Ситуация настолько распространенная, что про неё даже отдельный мем есть:","paragraph-yMj5CszTO",{"__TSPROSE_proseElement":219,"schema":576,"data":578,"storageKey":579,"id":581},{"name":577,"type":222,"linkable":219,"__TSPROSE_schema":219},"image",{"src":579,"width":580},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fa-b-squared-meme.svg","400px","image-xvwEH7CeL",{"__TSPROSE_proseElement":219,"schema":583,"children":584,"id":607},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[585,588,594,597,604],{"__TSPROSE_proseElement":219,"schema":586,"data":587},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Очень хотелось бы, чтобы последний ответ был именно таким. Так сильно хочется, что у этой мечты о возможности напрямую применить степень к слагаемым ",{"__TSPROSE_proseElement":219,"schema":589,"data":590,"storageKey":592,"id":593},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":591},"(a+b)^n = a^n + b^n","$ (a+b)^n = a^n + b^n $","inlinerMath-1TS0ztSM3",{"__TSPROSE_proseElement":219,"schema":595,"data":596},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," есть своё название -- ",{"__TSPROSE_proseElement":219,"schema":598,"data":600,"storageKey":602,"id":603},{"name":599,"type":238,"linkable":219,"__TSPROSE_schema":219},"referenceInliner",{"label":601},"\"Freshman's Dream\"","\u003Clink:external>\u002Fhttps:\u002F\u002Fw.wiki\u002FPjb","referenceInliner-Y2TDQc1D2",{"__TSPROSE_proseElement":219,"schema":605,"data":606},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", что можно перевести как \"Мечта первокурсника\". В несбыточности этой мечты можно убедиться очень быстро, попробовав подставить конкретные числа вместо букв:","paragraph-x728flO6B",{"__TSPROSE_proseElement":219,"schema":609,"data":610,"storageKey":612,"id":613},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":611,"freeze":223},"\\red{(1+2)^2 = 1^2 + 2^2 = 5} \\\\ \\boxed{\\green{(1 + 2)^2 = 3^2 = 9}} >>{big} \\red{(2+3)^3 = 2^3 + 3^3 = 35} \\\\ \\boxed{\\green{(2 + 3)^3 = 5^3 = 125}}","$$ \\red{(1+2)^2 = 1^2 + 2^2 = 5} \\\\ \\boxed{\\green{(1 + 2)^2 = 3^2 = 9}} >>{big} \\red{(2+3)^3 = 2^3 + 3^3 = 35} \\\\ \\boxed{\\green{(2 + 3)^3 = 5^3 = 125}} $$","blockMath-OQXfuW9pi",{"__TSPROSE_proseElement":219,"schema":615,"children":616,"id":670},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[617,620,626,629,639,642,648,651,660,663,667],{"__TSPROSE_proseElement":219,"schema":618,"data":619},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"В меме и первом примере степень равна двум, а выражение ",{"__TSPROSE_proseElement":219,"schema":621,"data":622,"storageKey":624,"id":625},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":623},"(a+b)^2","$ (a+b)^2 $","inlinerMath-K2ZDiwXLY",{"__TSPROSE_proseElement":219,"schema":627,"data":628},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," называют ",{"__TSPROSE_proseElement":219,"schema":630,"data":632,"children":634,"id":638},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},"emphasis",{"type":633},"bold",[635],{"__TSPROSE_proseElement":219,"schema":636,"data":637},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"\"квадратом суммы\"","emphasis-Ww55k8vfy",{"__TSPROSE_proseElement":219,"schema":640,"data":641},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", потому что ко всей этой сумме ",{"__TSPROSE_proseElement":219,"schema":643,"data":644,"storageKey":646,"id":647},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":645},"a + b","$ a + b $","inlinerMath-ZpaGPDgeL",{"__TSPROSE_proseElement":219,"schema":649,"data":650},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," применяется квадрат (возведение во вторую степень). Отсюда и название -- \"квадрат ",{"__TSPROSE_proseElement":219,"schema":652,"data":653,"children":655,"id":659},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":654},"italic",[656],{"__TSPROSE_proseElement":219,"schema":657,"data":658},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"всей","emphasis-giZhQaNj8",{"__TSPROSE_proseElement":219,"schema":661,"data":662},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," суммы\". Правильная формула для квадрата суммы ",{"__TSPROSE_proseElement":219,"schema":664,"data":665,"storageKey":624,"id":666},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":623},"inlinerMath-K2ZDiwXLY-1",{"__TSPROSE_proseElement":219,"schema":668,"data":669},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," очень похожа на \"Мечту первокурсника\", но чуть-чуть сложнее. Вывести её можно вручную и очень быстро. Самый прямой способ -- напрямую расписать квадрат как двойное произведение и умножить скобку на скобку методом \"фонтанчика\":","paragraph-Mc1VWWDrc",{"__TSPROSE_proseElement":219,"schema":672,"data":673,"storageKey":674,"children":677,"id":685},{"name":577,"type":222,"linkable":219,"__TSPROSE_schema":219},{"src":674,"width":675,"invert":676},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Ffountain-method.svg","300px","dark",[678],{"__TSPROSE_proseElement":219,"schema":679,"children":681},{"name":680,"type":238,"linkable":223,"__TSPROSE_schema":219},"caption",[682],{"__TSPROSE_proseElement":219,"schema":683,"data":684},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Умножение скобок методом \"фонтанчика\"","image-vdA2vlggV",{"__TSPROSE_proseElement":219,"schema":687,"data":688,"storageKey":690,"id":691},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":689,"freeze":223},"(a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \\boxed{a^2 + 2ab + b^2}","$$ (a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \\boxed{a^2 + 2ab + b^2} $$","blockMath-7lei3dGoV",{"__TSPROSE_proseElement":219,"schema":693,"children":694,"id":707},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[695,698,704],{"__TSPROSE_proseElement":219,"schema":696,"data":697},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"В обратную сторону, из \"суммы\" в запакованное \"произведение\", выводится тоже просто -- разбиваем ",{"__TSPROSE_proseElement":219,"schema":699,"data":700,"storageKey":702,"id":703},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":701},"2ab","$ 2ab $","inlinerMath-McBVEYBhA",{"__TSPROSE_proseElement":219,"schema":705,"data":706},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," на два слагаемых и несколько раз подряд производим вынесение общего множителя за скобки:","paragraph-5JgMD1cQv",{"__TSPROSE_proseElement":219,"schema":709,"data":710,"storageKey":712,"id":713},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":711,"freeze":223},"a^2 + 2ab + b^2 = a^2 + ab + ab + b^2 = a(a + b) + b(a + b) = (a + b)(a + b) = \\boxed{(a + b)^2}","$$ a^2 + 2ab + b^2 = a^2 + ab + ab + b^2 = a(a + b) + b(a + b) = (a + b)(a + b) = \\boxed{(a + b)^2} $$","blockMath-KeTVimB1e",{"__TSPROSE_proseElement":219,"schema":715,"children":716,"id":764},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[717,720,726,728,734,737,743,746,752,755,761],{"__TSPROSE_proseElement":219,"schema":718,"data":719},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Второй способ вывода -- геометрический. 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Не забывайте проводить сокращение!","paragraph-tRoi7xAAl",{"__TSPROSE_proseElement":219,"schema":978,"data":979,"storageKey":981,"id":982},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":980,"freeze":223},"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":219,"schema":984,"children":985,"id":988},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[986],{"__TSPROSE_proseElement":219,"schema":987,"data":904},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"paragraph-qkHzg2hJ6-1",{"__TSPROSE_proseElement":219,"schema":990,"data":991,"storageKey":993,"id":994},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":992,"freeze":223},"\\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":219,"schema":996,"data":997,"children":998},{"name":258,"type":222,"linkable":223,"__TSPROSE_schema":219},{},[999,1008,1014,1046,1054],{"__TSPROSE_proseElement":219,"schema":1000,"children":1001},{"name":263,"type":222,"linkable":223,"__TSPROSE_schema":219},[1002],{"__TSPROSE_proseElement":219,"schema":1003,"data":1004,"storageKey":1006,"id":1007},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1005,"freeze":223},"(-7a - 3b)^2","$$ (-7a - 3b)^2 $$","blockMath-FV48nxGyl",{"__TSPROSE_proseElement":219,"schema":1009,"data":1010},{"name":274,"type":222,"linkable":223,"__TSPROSE_schema":219},{"serializedValidator":1011},{"__ERUDIT_CHECK":219,"name":840,"data":1012},{"expr":1013},"49a^2 + 42ab + 9b^2",{"__TSPROSE_proseElement":219,"schema":1015,"children":1017},{"name":1016,"type":222,"linkable":223,"__TSPROSE_schema":219},"problemHint",[1018,1033,1039],{"__TSPROSE_proseElement":219,"schema":1019,"children":1020,"id":1032},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1021,1024,1029],{"__TSPROSE_proseElement":219,"schema":1022,"data":1023},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Вариант ",{"__TSPROSE_proseElement":219,"schema":1025,"data":1026,"storageKey":1027,"id":1028},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":389},"$ 1 $","inlinerMath-GWJ3eIxvK",{"__TSPROSE_proseElement":219,"schema":1030,"data":1031},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," - запишите сумму под квадратом с явным знаком плюса:","paragraph-WUD9q4TRq",{"__TSPROSE_proseElement":219,"schema":1034,"data":1035,"storageKey":1037,"id":1038},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1036,"freeze":223},"(-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2","$$ (-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2 $$","blockMath-YyTXKOh9m",{"__TSPROSE_proseElement":219,"schema":1040,"children":1041,"id":1045},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1042],{"__TSPROSE_proseElement":219,"schema":1043,"data":1044},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Вариант 2 -- внутри суммы вынесите минус за скобки.","paragraph-JvOAL3Gg5",{"__TSPROSE_proseElement":219,"schema":1047,"children":1048},{"name":282,"type":222,"linkable":223,"__TSPROSE_schema":219},[1049],{"__TSPROSE_proseElement":219,"schema":1050,"data":1051,"storageKey":1052,"id":1053},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1013,"freeze":223},"$$ 49a^2 + 42ab + 9b^2 $$","blockMath-JHMiXhAI7",{"__TSPROSE_proseElement":219,"schema":1055,"children":1056},{"name":291,"type":222,"linkable":223,"__TSPROSE_schema":219},[1057,1172],{"__TSPROSE_proseElement":219,"schema":1058,"data":1060,"children":1061},{"name":1059,"type":222,"linkable":223,"__TSPROSE_schema":219},"problemSection","Через явный знак плюса",[1062,1117,1121,1154,1160,1166],{"__TSPROSE_proseElement":219,"schema":1063,"children":1064,"id":1116},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1065,1068,1074,1077,1083,1085,1091,1094,1098,1101,1105,1107,1113],{"__TSPROSE_proseElement":219,"schema":1066,"data":1067},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Минус в сумме ",{"__TSPROSE_proseElement":219,"schema":1069,"data":1070,"storageKey":1072,"id":1073},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1071},"-7a - 3b","$ -7a - 3b $","inlinerMath-EOtl20nRc",{"__TSPROSE_proseElement":219,"schema":1075,"data":1076},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," можно рассматривать двояко. С одной стороны он может образовывать разность чисел ",{"__TSPROSE_proseElement":219,"schema":1078,"data":1079,"storageKey":1081,"id":1082},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1080},"-7a","$ -7a $","inlinerMath-80LHk8XHo",{"__TSPROSE_proseElement":219,"schema":1084,"data":334},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":1086,"data":1087,"storageKey":1089,"id":1090},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1088},"3b","$ 3b $","inlinerMath-OkRX6uD9W",{"__TSPROSE_proseElement":219,"schema":1092,"data":1093},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". Но можно этот минус \"прилепить\" к ",{"__TSPROSE_proseElement":219,"schema":1095,"data":1096,"storageKey":1089,"id":1097},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1088},"inlinerMath-OkRX6uD9W-1",{"__TSPROSE_proseElement":219,"schema":1099,"data":1100},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". И тогда получится сумма двух чисел ",{"__TSPROSE_proseElement":219,"schema":1102,"data":1103,"storageKey":1081,"id":1104},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1080},"inlinerMath-80LHk8XHo-1",{"__TSPROSE_proseElement":219,"schema":1106,"data":334},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":1108,"data":1109,"storageKey":1111,"id":1112},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1110},"-3b","$ -3b $","inlinerMath-Djaufz4Sl",{"__TSPROSE_proseElement":219,"schema":1114,"data":1115},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", которую явным образом можно записать вот так:","paragraph-WaBdeUTzI",{"__TSPROSE_proseElement":219,"schema":1118,"data":1119,"storageKey":1037,"id":1120},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1036,"freeze":223},"blockMath-YyTXKOh9m-1",{"__TSPROSE_proseElement":219,"schema":1122,"children":1123,"id":1153},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1124,1127,1131,1134,1138,1140,1144,1146,1150],{"__TSPROSE_proseElement":219,"schema":1125,"data":1126},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"А дальше по уже отработанной схеме. Роль ",{"__TSPROSE_proseElement":219,"schema":1128,"data":1129,"storageKey":724,"id":1130},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":723},"inlinerMath-YqDfaaRXA-3",{"__TSPROSE_proseElement":219,"schema":1132,"data":1133},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," в формуле квадрата суммы играет число ",{"__TSPROSE_proseElement":219,"schema":1135,"data":1136,"storageKey":1081,"id":1137},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1080},"inlinerMath-80LHk8XHo-2",{"__TSPROSE_proseElement":219,"schema":1139,"data":875},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":1141,"data":1142,"storageKey":732,"id":1143},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":731},"inlinerMath-LTS8nNQCZ-3",{"__TSPROSE_proseElement":219,"schema":1145,"data":882},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":1147,"data":1148,"storageKey":1111,"id":1149},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1110},"inlinerMath-Djaufz4Sl-1",{"__TSPROSE_proseElement":219,"schema":1151,"data":1152},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". Слева направо находим квадрат первого, удвоенное произведение первого на второе и квадрат второго.","paragraph-YgL94AVlu",{"__TSPROSE_proseElement":219,"schema":1155,"data":1156,"storageKey":1158,"id":1159},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1157,"freeze":223},"(-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":219,"schema":1161,"children":1162,"id":1165},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1163],{"__TSPROSE_proseElement":219,"schema":1164,"data":904},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"paragraph-qkHzg2hJ6-2",{"__TSPROSE_proseElement":219,"schema":1167,"data":1168,"storageKey":1170,"id":1171},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1169,"freeze":223},"(-7a - 3b)^2 = 49a^2 + 42ab + 9b^2","$$ (-7a - 3b)^2 = 49a^2 + 42ab + 9b^2 $$","blockMath-A2ZrMc5jo",{"__TSPROSE_proseElement":219,"schema":1173,"data":1174,"children":1175},{"name":1059,"type":222,"linkable":223,"__TSPROSE_schema":219},"Вынесение минуса за скобки",[1176,1190,1196,1203],{"__TSPROSE_proseElement":219,"schema":1177,"children":1178,"id":1189},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1179,1182,1186],{"__TSPROSE_proseElement":219,"schema":1180,"data":1181},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Вынесем минус за скобки из выражения ",{"__TSPROSE_proseElement":219,"schema":1183,"data":1184,"storageKey":1072,"id":1185},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":1071},"inlinerMath-EOtl20nRc-1",{"__TSPROSE_proseElement":219,"schema":1187,"data":1188},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},":","paragraph-8ANRciv58",{"__TSPROSE_proseElement":219,"schema":1191,"data":1192,"storageKey":1194,"id":1195},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1193,"freeze":223},"-7a - 3b = -(7a + 3b)","$$ -7a - 3b = -(7a + 3b) $$","blockMath-HfGK3huoD",{"__TSPROSE_proseElement":219,"schema":1197,"children":1198,"id":1202},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1199],{"__TSPROSE_proseElement":219,"schema":1200,"data":1201},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Теперь у нас есть сумма внутри скобок, а отдельный минус \"уничтожит\" возведение в квадрат:","paragraph-6UJJ48xBQ",{"__TSPROSE_proseElement":219,"schema":1204,"data":1205,"storageKey":1207,"id":1208},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1206,"freeze":223},"(-(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","square-sum-expand-examples",{"__TSPROSE_proseElement":219,"schema":1211,"children":1212,"id":1216},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1213],{"__TSPROSE_proseElement":219,"schema":1214,"data":1215},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Формула квадрата суммы позволяет не только быстро раскрывать скобки, но и наоборот -- запаковывать уже разложенные выражения обратно в \"скобку в квадрате\", в квадрат суммы. Этот процесс часто называют \"выделением полного квадрата\" и о нем мы еще поговорим отдельно ниже.","paragraph-jB8SVbmal",{"__TSPROSE_proseElement":219,"schema":1218,"children":1219,"id":1274},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[1220,1223,1227,1229,1233,1236,1240,1243,1251,1254,1258,1260,1264,1267,1271],{"__TSPROSE_proseElement":219,"schema":1221,"data":1222},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Процесс \"запаковки\" чуть сложнее разложения. Основная цель -- найти ",{"__TSPROSE_proseElement":219,"schema":1224,"data":1225,"storageKey":724,"id":1226},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":723},"inlinerMath-YqDfaaRXA-4",{"__TSPROSE_proseElement":219,"schema":1228,"data":334},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":1230,"data":1231,"storageKey":732,"id":1232},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":731},"inlinerMath-LTS8nNQCZ-4",{"__TSPROSE_proseElement":219,"schema":1234,"data":1235},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", чтобы составить квадрат суммы ",{"__TSPROSE_proseElement":219,"schema":1237,"data":1238,"storageKey":624,"id":1239},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":623},"inlinerMath-K2ZDiwXLY-2",{"__TSPROSE_proseElement":219,"schema":1241,"data":1242},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". 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Первый и самый быстрый -- посмотреть на ",{"__TSPROSE_proseElement":219,"schema":1244,"data":1245,"children":1246,"id":1250},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[1247],{"__TSPROSE_proseElement":219,"schema":1248,"data":1249},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"крайние слагаемые","emphasis-OQVpMgRTM",{"__TSPROSE_proseElement":219,"schema":1252,"data":1253},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},": в простых случаях там сразу видны \"красивые\" квадраты, которые мгновенно выдают ",{"__TSPROSE_proseElement":219,"schema":1255,"data":1256,"storageKey":724,"id":1257},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":723},"inlinerMath-YqDfaaRXA-5",{"__TSPROSE_proseElement":219,"schema":1259,"data":334},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":1261,"data":1262,"storageKey":732,"id":1263},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":731},"inlinerMath-LTS8nNQCZ-5",{"__TSPROSE_proseElement":219,"schema":1265,"data":1266},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". 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$","inlinerMath-VR8AvPckN",{"__TSPROSE_proseElement":219,"schema":2062,"data":1692},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"paragraph-DpwMVRKAJ",{"__TSPROSE_proseElement":219,"schema":2065,"data":2066,"storageKey":1979,"id":2067},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":1978,"freeze":223},"blockMath-akHjm2BVw-1","square-sum-factor-examples",{"__TSPROSE_proseElement":219,"schema":2070,"data":2072,"storageKey":2073,"children":2075,"id":2112},{"name":2071,"type":222,"linkable":219,"__TSPROSE_schema":219},"callout",{"iconSrc":2073,"title":2074},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fancient-formula.webp","Очень древние формулы",[2076,2083,2096],{"__TSPROSE_proseElement":219,"schema":2077,"children":2078,"id":2082},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2079],{"__TSPROSE_proseElement":219,"schema":2080,"data":2081},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Формулы сокращенного умножения были известны еще в глубокой древности, аж древнегреческим математикам. Например Евклиду, который уже в 3-ем веке до нашей эры использовал их геометрически для подсчета площадей. Формулу квадрата суммы он сформулировал вот так:","paragraph-VPwgB7uSh",{"__TSPROSE_proseElement":219,"schema":2084,"data":2085,"children":2086,"id":2095},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},{"center":219},[2087],{"__TSPROSE_proseElement":219,"schema":2088,"data":2089,"children":2090,"id":2094},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":654},[2091],{"__TSPROSE_proseElement":219,"schema":2092,"data":2093},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"\"Если отрезок как-либо разбит на два отрезка, то площадь квадрата, построенного на всём отрезке, равна сумме площадей квадратов, построенных на каждом из отрезков, и удвоенной площади прямоугольника, сторонами которого служат эти два отрезка.\"","emphasis-aYJH0orbk","paragraph-p30e1hKYH",{"__TSPROSE_proseElement":219,"schema":2097,"children":2098,"id":2111},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2099,2102,2108],{"__TSPROSE_proseElement":219,"schema":2100,"data":2101},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Современный вид ФСУ приобрели много позже, в 16-17 веках, благодаря математикам Франсуа Виету (тот самый, в честь которого названы ",{"__TSPROSE_proseElement":219,"schema":2103,"data":2104,"storageKey":2106,"id":2107},{"name":599,"type":238,"linkable":219,"__TSPROSE_schema":219},{"label":2105},"формулы Виета","\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas","referenceInliner-vhgUJfqU3",{"__TSPROSE_proseElement":219,"schema":2109,"data":2110},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},") и Рене Декарту. До сих пор эти формулы являются одними из самых часто используемых математических \"трюков\" везде в математике, от упрощения выражений в алгебре до разбития уравнений на множители в криптографии.","paragraph-UuLxoUjL4","callout-EIdpSurZ2",{"__TSPROSE_proseElement":219,"schema":2114,"data":2115,"id":2117},{"name":227,"type":222,"linkable":219,"__TSPROSE_schema":219},{"level":172,"title":2116},"Квадрат разности","kvadrat-raznosti",{"__TSPROSE_proseElement":219,"schema":2119,"children":2120,"id":2152},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2121,2124,2130,2133,2139,2142,2149],{"__TSPROSE_proseElement":219,"schema":2122,"data":2123},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"По аналогии с квадратом суммы, \"квадратом разности\" называют выражение вида ",{"__TSPROSE_proseElement":219,"schema":2125,"data":2126,"storageKey":2128,"id":2129},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":2127},"(a-b)^2","$ (a-b)^2 $","inlinerMath-kgXTO8hrU",{"__TSPROSE_proseElement":219,"schema":2131,"data":2132},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". И понятно почему, ведь у нас есть некая разность между двумя числами ",{"__TSPROSE_proseElement":219,"schema":2134,"data":2135,"storageKey":2137,"id":2138},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":2136},"a-b","$ a-b $","inlinerMath-zxME95g2R",{"__TSPROSE_proseElement":219,"schema":2140,"data":2141},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," и мы хотим возвести всю эту разность целиком в квадрат, то есть во вторую степень. Отсюда и получается название -- \"квадрат ",{"__TSPROSE_proseElement":219,"schema":2143,"data":2144,"children":2145,"id":2148},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":654},[2146],{"__TSPROSE_proseElement":219,"schema":2147,"data":658},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"emphasis-giZhQaNj8-1",{"__TSPROSE_proseElement":219,"schema":2150,"data":2151},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," разности\".","paragraph-jt6Ad7iEr",{"__TSPROSE_proseElement":219,"schema":2154,"children":2155,"id":2159},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2156],{"__TSPROSE_proseElement":219,"schema":2157,"data":2158},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Алгеброически формула квадрата разности в \"обе стороны\" выводится ровно точно так же, как и формула квадрата суммы: раскрываем скобки фонтанчиком в одну сторону или разбиваем удвоенное слагаемое с выносом за скобки общих множителей в другую:","paragraph-QgRdc8DRc",{"__TSPROSE_proseElement":219,"schema":2161,"data":2162,"storageKey":2164,"id":2165},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":2163,"freeze":223},"(a - b)^2 = (a - b)(a - b) = a^2 - ab - ba + b^2 = \\boxed{a^2 - 2ab + b^2} \\\\ a^2 - 2ab + b^2 = a^2 - ab - ab + b^2 = a(a - b) - b(a - b) = (a - b)(a - b) = \\boxed{(a - b)^2}","$$ (a - b)^2 = (a - b)(a - b) = a^2 - ab - ba + b^2 = \\boxed{a^2 - 2ab + b^2} \\\\ a^2 - 2ab + b^2 = a^2 - ab - ab + b^2 = a(a - b) - b(a - b) = (a - b)(a - b) = \\boxed{(a - b)^2} $$","blockMath-4ExN6IPzS",{"__TSPROSE_proseElement":219,"schema":2167,"children":2168,"id":2192},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2169,2172,2176,2179,2183,2186,2190],{"__TSPROSE_proseElement":219,"schema":2170,"data":2171},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Геометрический вывод тоже возможен. В квадрате суммы мы искали общую площадь большого квадрата со стороной из двух отрезков. Сейчас у нас уже есть большой квадрат со стороной ",{"__TSPROSE_proseElement":219,"schema":2173,"data":2174,"storageKey":724,"id":2175},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":723},"inlinerMath-YqDfaaRXA-8",{"__TSPROSE_proseElement":219,"schema":2177,"data":2178},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", и мы укорачиваем его стороны на длину ",{"__TSPROSE_proseElement":219,"schema":2180,"data":2181,"storageKey":732,"id":2182},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":731},"inlinerMath-LTS8nNQCZ-8",{"__TSPROSE_proseElement":219,"schema":2184,"data":2185},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". Площадь уменьшенного квадрата и будет численно равна ",{"__TSPROSE_proseElement":219,"schema":2187,"data":2188,"storageKey":2128,"id":2189},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":2127},"inlinerMath-kgXTO8hrU-1",{"__TSPROSE_proseElement":219,"schema":2191,"data":1188},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"paragraph-0iLlk8j56",{"__TSPROSE_proseElement":219,"schema":2194,"data":2195,"storageKey":2196,"children":2197,"id":2204},{"name":577,"type":222,"linkable":219,"__TSPROSE_schema":219},{"src":2196,"invert":676},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-diff-schema.svg",[2198],{"__TSPROSE_proseElement":219,"schema":2199,"children":2200},{"name":680,"type":238,"linkable":223,"__TSPROSE_schema":219},[2201],{"__TSPROSE_proseElement":219,"schema":2202,"data":2203},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Геометрический вывод формулы квадрата разности","image-ochFeUxTg",{"__TSPROSE_proseElement":219,"schema":2206,"data":2207,"children":2208,"id":2218},{"name":188,"type":222,"linkable":219,"__TSPROSE_schema":219},{"title":191,"layout":540},[2209],{"__TSPROSE_proseElement":219,"schema":2210,"children":2211},{"name":783,"type":222,"linkable":223,"__TSPROSE_schema":219},[2212],{"__TSPROSE_proseElement":219,"schema":2213,"data":2214,"storageKey":2216,"id":2217},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":2215,"freeze":223},"(a - b)^2 = a^2 - 2ab + b^2","$$ (a - b)^2 = a^2 - 2ab + b^2 $$","blockMath-xpqkz7Ecb","square-diff",{"__TSPROSE_proseElement":219,"schema":2220,"children":2221,"id":2225},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2222],{"__TSPROSE_proseElement":219,"schema":2223,"data":2224},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Как видите, формула квадрата разности отличается от формулы квадрата суммы только изменением первого знака с плюса на минус. Достаточно помнить \"плюсовой\" её вариант и при необходимости первый встречный знак поменять на минус.","paragraph-Zt4w5tcIj",{"__TSPROSE_proseElement":219,"schema":2227,"data":2229,"children":2231,"id":2296},{"name":2228,"type":222,"linkable":219,"__TSPROSE_schema":219},"accent_important",{"title":2230,"layout":540},"Минус есть, а все равно сумма!",[2232],{"__TSPROSE_proseElement":219,"schema":2233,"children":2235},{"name":2234,"type":222,"linkable":223,"__TSPROSE_schema":219},"accentMain_important",[2236,2272,2278],{"__TSPROSE_proseElement":219,"schema":2237,"children":2238,"id":2271},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2239,2242,2250,2253,2259,2262,2268],{"__TSPROSE_proseElement":219,"schema":2240,"data":2241},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Из-за двоякости знака \"минус\" в математике ",{"__TSPROSE_proseElement":219,"schema":2243,"data":2244,"children":2245,"id":2249},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633,"accent":219},[2246],{"__TSPROSE_proseElement":219,"schema":2247,"data":2248},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"любую \"разность\" можно представить как сумму","emphasis-QAbHC6HxG",{"__TSPROSE_proseElement":219,"schema":2251,"data":2252},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", если под знаками минуса понимать отрицание числа. Не \"три минус два\" ",{"__TSPROSE_proseElement":219,"schema":2254,"data":2255,"storageKey":2257,"id":2258},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":2256},"3-2","$ 3-2 $","inlinerMath-D102lV2Oh",{"__TSPROSE_proseElement":219,"schema":2260,"data":2261},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", а \"сложение тройки с отрицательной двойкой\" ",{"__TSPROSE_proseElement":219,"schema":2263,"data":2264,"storageKey":2266,"id":2267},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":2265},"3+(-2)","$ 3+(-2) $","inlinerMath-rXqEDLuYB",{"__TSPROSE_proseElement":219,"schema":2269,"data":2270},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". С выражениями тоже работает:","paragraph-CuzX0ordC",{"__TSPROSE_proseElement":219,"schema":2273,"data":2274,"storageKey":2276,"id":2277},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":2275,"freeze":223},"- a - b + c - d = (-a) + (-b) + c + (-d)","$$ - a - b + c - d = (-a) + (-b) + c + (-d) $$","blockMath-bBmMzTW6H",{"__TSPROSE_proseElement":219,"schema":2279,"children":2280,"id":2295},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2281,2284,2292],{"__TSPROSE_proseElement":219,"schema":2282,"data":2283},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Поэтому не удивляйтесь, когда математики называют какие-то выражения \"суммами\", ",{"__TSPROSE_proseElement":219,"schema":2285,"data":2286,"children":2287,"id":2291},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[2288],{"__TSPROSE_proseElement":219,"schema":2289,"data":2290},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"даже если никаких плюсов там нет!","emphasis-Ddm2HNSKX",{"__TSPROSE_proseElement":219,"schema":2293,"data":2294},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," Мы будем поступать точно так же.","paragraph-y6hPUihXQ","minus-est-a-vse-ravno-summa",{"__TSPROSE_proseElement":219,"schema":2298,"children":2299,"id":2319},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[2300,2303,2307,2310,2316],{"__TSPROSE_proseElement":219,"schema":2301,"data":2302},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Потренируйтесь с помощью формулы квадрата разности быстро раскрывать скобки и запаковывать выражения обратно. Схема использования этой формулы точно такая, как и квадрата суммы, главное в минусах не запутайтесь. При \"запаковке\" сумм в скобки делить надо не на ",{"__TSPROSE_proseElement":219,"schema":2304,"data":2305,"storageKey":529,"id":2306},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":528},"inlinerMath-z9Yar9waf-7",{"__TSPROSE_proseElement":219,"schema":2308,"data":2309},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", а на ",{"__TSPROSE_proseElement":219,"schema":2311,"data":2312,"storageKey":2314,"id":2315},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":2313},"-2","$ -2 $","inlinerMath-02bjIi1Ry",{"__TSPROSE_proseElement":219,"schema":2317,"data":2318},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". 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Это позволяет упрощать сложные выражения. 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Для этого из большего квадрата площадью ",{"__TSPROSE_proseElement":219,"schema":4084,"data":4085,"storageKey":741,"id":4086},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":740},"inlinerMath-eiXlIGTfO-1",{"__TSPROSE_proseElement":219,"schema":4088,"data":4089},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," вырежем меньший квадрат площадью ",{"__TSPROSE_proseElement":219,"schema":4091,"data":4092,"storageKey":759,"id":4093},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":758},"inlinerMath-tBkcHvoEV-1",{"__TSPROSE_proseElement":219,"schema":4095,"data":4096},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". 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Для этого надо просто раскрыть эти скобки любым образом, например по \"правилу фонтанчика\":","paragraph-TN607RmYt",{"__TSPROSE_proseElement":219,"schema":4162,"data":4163,"storageKey":4165,"id":4166},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":4164,"freeze":223},"(a+b)(a-b) = a \\cdot a - a \\cdot b + b \\cdot a - b \\cdot b = \\boxed{a^2 - b^2}","$$ (a+b)(a-b) = a \\cdot a - a \\cdot b + b \\cdot a - b \\cdot b = \\boxed{a^2 - b^2} $$","blockMath-wMfvEsRpC",{"__TSPROSE_proseElement":219,"schema":4168},{"name":4169,"type":222,"linkable":223,"__TSPROSE_schema":219},"hr",{"__TSPROSE_proseElement":219,"schema":4171,"children":4172,"id":4176},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[4173],{"__TSPROSE_proseElement":219,"schema":4174,"data":4175},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"В обратную сторону выводится чуть сложнее. Нам нужно искусственно добавить и сразу вычесть слагаемое, а затем выносить общие множители за скобки:","paragraph-RjuWFPIC7",{"__TSPROSE_proseElement":219,"schema":4178,"data":4179,"storageKey":4181,"id":4182},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":4180,"freeze":223},"a^2 - b^2 = a^2 + \\underbrace{ab - ab}_{\\text{Добавили и вычли}} - b^2 = a(a+b) - b(a+b) = \\boxed{(a+b)(a-b)}","$$ a^2 - b^2 = a^2 + \\underbrace{ab - ab}_{\\text{Добавили и вычли}} - b^2 = a(a+b) - b(a+b) = \\boxed{(a+b)(a-b)} $$","blockMath-IeRVWM9rK","diff-of-squares",{"__TSPROSE_proseElement":219,"schema":4185,"data":4186,"children":4188,"id":4257},{"name":2228,"type":222,"linkable":219,"__TSPROSE_schema":219},{"title":4187,"layout":540},"Разность квадратов ≠ Квадрат разности!",[4189],{"__TSPROSE_proseElement":219,"schema":4190,"children":4191},{"name":2234,"type":222,"linkable":223,"__TSPROSE_schema":219},[4192],{"__TSPROSE_proseElement":219,"schema":4193,"children":4194,"id":4256},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[4195,4198,4206,4209,4217,4220,4224,4227,4235,4238,4246,4248,4254],{"__TSPROSE_proseElement":219,"schema":4196,"data":4197},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Изучающие ФСУ новички очень часто путают разность квадратов с квадратом разности. 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А вся её фишка в том, что она позволяет практически на коленке вдвое снизить или повысить степень двух любых выражений. Скоро вы сами убедитесь, сколько всего эта одна простая формула резко упрощает и как лихо она \"сворачивает\" сложные выражения. 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И получили аж три интересные формулы? квадрат суммы, квадрат разности и разность квадратов. А можно ли пойти дальше? Возможно, существуют удобные формулы для быстрой работы с кубами, то есть с третьей степенью? Конечно же они есть! Учить наизусть не обязательно, но полезно будет хотя бы ознакомиться с ними.","paragraph-vjUEDJCGr",{"__TSPROSE_proseElement":219,"schema":4681,"children":4682,"id":4686},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[4683],{"__TSPROSE_proseElement":219,"schema":4684,"data":4685},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Вообще ФСУ довольно много, но мы затроним только самые базовые -- куб суммы и разности. Названия говорят сами за себя. Куб суммы\u002Fразности означает, что есть какая-то сумма или разность двух чисел, и вся она взята в куб, то есть в третью степень. Формулы для них выводятся так же, как и для квадрата суммы -- раскрытием скобок по \"методу фонтанчика\":","paragraph-jNQ8n92q1",{"__TSPROSE_proseElement":219,"schema":4688,"data":4689,"storageKey":4691,"id":4692},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":4690,"freeze":223},"(a+b)^3 = (a+b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = \\boxed{a^3 + 3a^2b + 3ab^2 + b^3} \\\\ (a-b)^3 = (a-b)^2(a-b) = (a^2 - 2ab + b^2)(a-b) = \\boxed{a^3 - 3a^2b + 3ab^2 - b^3}","$$ (a+b)^3 = (a+b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = \\boxed{a^3 + 3a^2b + 3ab^2 + b^3} \\\\ (a-b)^3 = (a-b)^2(a-b) = (a^2 - 2ab + b^2)(a-b) = \\boxed{a^3 - 3a^2b + 3ab^2 - b^3} $$","blockMath-7jaM7Hka9",{"__TSPROSE_proseElement":219,"schema":4694,"data":4695,"storageKey":4697,"id":4698},{"name":4033,"type":222,"linkable":219,"__TSPROSE_schema":219},{"label":4696},"Вывод в обратную сторону, из разложенной формы в запакованную, уже довольно хитрый. Настолько, что он вполне тянет на отдельную задачку вам на подумать 😈 Можете решить её сейчас или отложить до момента, пока не добьете статью.","\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fpractice\u002F$cubeSumDiffFactorization","referenceBlock-RDwczHiAz",{"__TSPROSE_proseElement":219,"schema":4700,"children":4701,"id":4705},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[4702],{"__TSPROSE_proseElement":219,"schema":4703,"data":4704},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Геометрически обе формулы тоже можно вывести. Уже из названия можно понять, что речь будет идти уже о трехмерных фигурах. И собирать мы будем не квадрат, а куб. Выглядит это вот так:","paragraph-CWP3kcbnC",{"__TSPROSE_proseElement":219,"schema":4707,"data":4709,"storageKey":4710,"children":4713,"id":4733},{"name":4708,"type":222,"linkable":219,"__TSPROSE_schema":219},"video",{"src":4710,"autoplay":219,"width":4711,"invert":4712},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fcube-sum.mp4","380px","light",[4714],{"__TSPROSE_proseElement":219,"schema":4715,"children":4716},{"name":680,"type":238,"linkable":223,"__TSPROSE_schema":219},[4717,4720],{"__TSPROSE_proseElement":219,"schema":4718,"data":4719},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Геометрический вывод куба суммы",{"__TSPROSE_proseElement":219,"schema":4721,"children":4723},{"name":4722,"type":238,"linkable":223,"__TSPROSE_schema":219},"captionSecondary",[4724,4727],{"__TSPROSE_proseElement":219,"schema":4725,"data":4726},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Взято с TikTok канала ",{"__TSPROSE_proseElement":219,"schema":4728,"data":4729,"storageKey":4731,"id":4732},{"name":599,"type":238,"linkable":219,"__TSPROSE_schema":219},{"label":4730},"@complex_math","\u003Clink:external>\u002Fhttps:\u002F\u002Fwww.tiktok.com\u002F@complex_math\u002Fvideo\u002F7358570064724970759","referenceInliner-cQNN2wVfA","video-7BY2L54CV",{"__TSPROSE_proseElement":219,"schema":4735,"children":4736,"id":4793},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[4737,4740,4746,4749,4755,4758,4764,4767,4771,4773,4777,4780,4784,4786,4790],{"__TSPROSE_proseElement":219,"schema":4738,"data":4739},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Из этой визуализации сразу становится понятно, откуда в формуле коэффициенты ",{"__TSPROSE_proseElement":219,"schema":4741,"data":4742,"storageKey":4744,"id":4745},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":4743},"3","$ 3 $","inlinerMath-mqn34wvB0",{"__TSPROSE_proseElement":219,"schema":4747,"data":4748},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," и выражения вида ",{"__TSPROSE_proseElement":219,"schema":4750,"data":4751,"storageKey":4753,"id":4754},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":4752},"a^2b","$ a^2b $","inlinerMath-GU9lQOdXE",{"__TSPROSE_proseElement":219,"schema":4756,"data":4757},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," ",{"__TSPROSE_proseElement":219,"schema":4759,"data":4760,"storageKey":4762,"id":4763},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":4761},"ab^2","$ ab^2 $","inlinerMath-U6dXU6jrQ",{"__TSPROSE_proseElement":219,"schema":4765,"data":4766},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". 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Приведем каждый член к виду, в котором будет видно, что он соответствует формуле куба суммы, подпишем их и запакуем:","paragraph-KoTbwAXF6",{"__TSPROSE_proseElement":219,"schema":5074,"data":5075,"storageKey":5077,"id":5078},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5076,"freeze":223},"\\underbrace{m^3}_{a^3} + 3 \\cdot \\underbrace{m^2}_{a^2} \\cdot \\underbrace{2n}_{b} + 3 \\cdot \\underbrace{m}_{a} \\cdot \\underbrace{(2n)^2}_{b^2} + \\underbrace{(2n)^3}_{b^3} = (m + 2n)^3","$$ \\underbrace{m^3}_{a^3} + 3 \\cdot \\underbrace{m^2}_{a^2} \\cdot \\underbrace{2n}_{b} + 3 \\cdot \\underbrace{m}_{a} \\cdot \\underbrace{(2n)^2}_{b^2} + \\underbrace{(2n)^3}_{b^3} = (m + 2n)^3 $$","blockMath-nD4M5tIIP",{"__TSPROSE_proseElement":219,"schema":5080,"data":5081,"children":5082},{"name":258,"type":222,"linkable":223,"__TSPROSE_schema":219},{},[5083,5098,5104,5113],{"__TSPROSE_proseElement":219,"schema":5084,"children":5085},{"name":263,"type":222,"linkable":223,"__TSPROSE_schema":219},[5086,5092],{"__TSPROSE_proseElement":219,"schema":5087,"children":5088,"id":5091},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5089],{"__TSPROSE_proseElement":219,"schema":5090,"data":4283},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"paragraph-U23zaEl70-6",{"__TSPROSE_proseElement":219,"schema":5093,"data":5094,"storageKey":5096,"id":5097},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5095,"freeze":223},"\\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27}","$$ \\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} $$","blockMath-AGVrgitHm",{"__TSPROSE_proseElement":219,"schema":5099,"data":5100},{"name":274,"type":222,"linkable":223,"__TSPROSE_schema":219},{"serializedValidator":5101},{"__ERUDIT_CHECK":219,"name":840,"data":5102},{"expr":5103},"(x\u002F2 - y\u002F3)^3",{"__TSPROSE_proseElement":219,"schema":5105,"children":5106},{"name":282,"type":222,"linkable":223,"__TSPROSE_schema":219},[5107],{"__TSPROSE_proseElement":219,"schema":5108,"data":5109,"storageKey":5111,"id":5112},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5110,"freeze":223},"\\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3","$$ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3 $$","blockMath-bBCuNpUCR",{"__TSPROSE_proseElement":219,"schema":5114,"children":5115},{"name":291,"type":222,"linkable":223,"__TSPROSE_schema":219},[5116,5167],{"__TSPROSE_proseElement":219,"schema":5117,"children":5118,"id":5166},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5119,5121,5125,5127,5131,5133,5139,5141,5147,5149,5155,5157,5163],{"__TSPROSE_proseElement":219,"schema":5120,"data":5024},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":5122,"data":5123,"storageKey":724,"id":5124},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":723},"inlinerMath-YqDfaaRXA-20",{"__TSPROSE_proseElement":219,"schema":5126,"data":334},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":5128,"data":5129,"storageKey":732,"id":5130},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":731},"inlinerMath-LTS8nNQCZ-20",{"__TSPROSE_proseElement":219,"schema":5132,"data":5037},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":5134,"data":5135,"storageKey":5137,"id":5138},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5136},"a^3 = x^3\u002F8 = (x\u002F2)^3","$ a^3 = x^3\u002F8 = (x\u002F2)^3 $","inlinerMath-YK5vQm7F8",{"__TSPROSE_proseElement":219,"schema":5140,"data":1496},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":5142,"data":5143,"storageKey":5145,"id":5146},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5144},"a = x\u002F2","$ a = x\u002F2 $","inlinerMath-6CbpESr7U",{"__TSPROSE_proseElement":219,"schema":5148,"data":5054},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":5150,"data":5151,"storageKey":5153,"id":5154},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5152},"b^3 = y^3\u002F27 = (y\u002F3)^3","$ b^3 = y^3\u002F27 = (y\u002F3)^3 $","inlinerMath-D7ku1JQ0R",{"__TSPROSE_proseElement":219,"schema":5156,"data":1496},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":5158,"data":5159,"storageKey":5161,"id":5162},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5160},"b = y\u002F3","$ b = y\u002F3 $","inlinerMath-3xOiTfiAj",{"__TSPROSE_proseElement":219,"schema":5164,"data":5165},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". Приведем каждый член к виду, в котором будет видно, что он соответствует формуле куба разности, подпишем их и запакуем:","paragraph-nNgyn3uYz",{"__TSPROSE_proseElement":219,"schema":5168,"data":5169,"storageKey":5171,"id":5172},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5170,"freeze":223},"\\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} = \\\\ \\underbrace{\\frac{x^3}{8}}_{a^3} - 3 \\cdot \\underbrace{\\frac{x^2}{4}}_{a^2} \\cdot \\underbrace{\\frac{y}{3}}_{b} + 3 \\cdot \\underbrace{\\frac{x}{2}}_{a} \\cdot \\underbrace{\\frac{y^2}{9}}_{b^2} - \\underbrace{\\frac{y^3}{27}}_{b^3} = \\\\ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3","$$ \\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} = \\\\ \\underbrace{\\frac{x^3}{8}}_{a^3} - 3 \\cdot \\underbrace{\\frac{x^2}{4}}_{a^2} \\cdot \\underbrace{\\frac{y}{3}}_{b} + 3 \\cdot \\underbrace{\\frac{x}{2}}_{a} \\cdot \\underbrace{\\frac{y^2}{9}}_{b^2} - \\underbrace{\\frac{y^3}{27}}_{b^3} = \\\\ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3 $$","blockMath-4EhtSSJbY","cube-sum-diff-examples",{"__TSPROSE_proseElement":219,"schema":5175,"data":5176,"id":5178},{"name":227,"type":222,"linkable":219,"__TSPROSE_schema":219},{"level":172,"title":5177},"Как запомнить ФСУ?","kak-zapomnit-fsu",{"__TSPROSE_proseElement":219,"schema":5180,"children":5181,"id":5208},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5182,5185,5191,5194,5198,5201,5205],{"__TSPROSE_proseElement":219,"schema":5183,"data":5184},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Наизусть запомнить нужно только три формулы: квадрат суммы ",{"__TSPROSE_proseElement":219,"schema":5186,"data":5187,"storageKey":5189,"id":5190},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5188},"(a + b)^2","$ (a + b)^2 $","inlinerMath-ECN7SKZxy",{"__TSPROSE_proseElement":219,"schema":5192,"data":5193},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", квадрат разности ",{"__TSPROSE_proseElement":219,"schema":5195,"data":5196,"storageKey":4252,"id":5197},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":4251},"inlinerMath-sAdriU9tc-1",{"__TSPROSE_proseElement":219,"schema":5199,"data":5200},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," и разность квадратов ",{"__TSPROSE_proseElement":219,"schema":5202,"data":5203,"storageKey":4053,"id":5204},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":4052},"inlinerMath-T1YOXnC6X-3",{"__TSPROSE_proseElement":219,"schema":5206,"data":5207},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". Они просто вездесущие и абсолютно необходимо уметь их сразу замечать и заменять разложение на скобки или наоброт. А вот формулы с кубами желательно просто уметь узнавать. Вот несколько советов, как проще все эти формулы запоминать:","paragraph-VZQUDPSIh",{"__TSPROSE_proseElement":219,"schema":5210,"data":5212,"children":5214,"id":5418},{"name":5211,"type":222,"linkable":219,"__TSPROSE_schema":219},"list",{"type":5213},"unordered",[5215,5289,5348],{"__TSPROSE_proseElement":219,"schema":5216,"children":5218},{"name":5217,"type":222,"linkable":223,"__TSPROSE_schema":219},"listItem",[5219,5231,5254,5260],{"__TSPROSE_proseElement":219,"schema":5220,"children":5221,"id":5230},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5222],{"__TSPROSE_proseElement":219,"schema":5223,"data":5224,"children":5225,"id":5229},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[5226],{"__TSPROSE_proseElement":219,"schema":5227,"data":5228},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Знак плюс-минус","emphasis-1T20LvUmV","paragraph-1JIYzh2eP",{"__TSPROSE_proseElement":219,"schema":5232,"children":5233,"id":5253},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5234,5237,5243,5246,5250],{"__TSPROSE_proseElement":219,"schema":5235,"data":5236},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Не запоминайте по отдельности ",{"__TSPROSE_proseElement":219,"schema":5238,"data":5239,"storageKey":5241,"id":5242},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5240},"4","$ 4 $","inlinerMath-ZXyaHjumL",{"__TSPROSE_proseElement":219,"schema":5244,"data":5245},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," формулы: квадрат суммы, квадрат разности, куб суммы и куб разности. Достаточно запомнить ",{"__TSPROSE_proseElement":219,"schema":5247,"data":5248,"storageKey":529,"id":5249},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":528},"inlinerMath-z9Yar9waf-12",{"__TSPROSE_proseElement":219,"schema":5251,"data":5252},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," формулы, если использовать знак плюс-минус, ведь кроме знаков больше ничего в них не меняется:","paragraph-TP6k3DSmh",{"__TSPROSE_proseElement":219,"schema":5255,"data":5256,"storageKey":5258,"id":5259},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5257,"freeze":223},"(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":219,"schema":5261,"children":5262,"id":5288},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5263,5266,5274,5277,5285],{"__TSPROSE_proseElement":219,"schema":5264,"data":5265},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"В суммах все знаки ",{"__TSPROSE_proseElement":219,"schema":5267,"data":5268,"children":5269,"id":5273},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[5270],{"__TSPROSE_proseElement":219,"schema":5271,"data":5272},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"всегда плюсы","emphasis-WATBgUbHA",{"__TSPROSE_proseElement":219,"schema":5275,"data":5276},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},". В разности ",{"__TSPROSE_proseElement":219,"schema":5278,"data":5279,"children":5280,"id":5284},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[5281],{"__TSPROSE_proseElement":219,"schema":5282,"data":5283},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"минус всегда идет сразу","emphasis-DzWKDXE2b",{"__TSPROSE_proseElement":219,"schema":5286,"data":5287},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," после первого слагаемого в разложении. В случае куба еще и у последнего слагаемого.","paragraph-NJkC7n6aH",{"__TSPROSE_proseElement":219,"schema":5290,"children":5291},{"name":5217,"type":222,"linkable":223,"__TSPROSE_schema":219},[5292,5304,5317],{"__TSPROSE_proseElement":219,"schema":5293,"children":5294,"id":5303},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5295],{"__TSPROSE_proseElement":219,"schema":5296,"data":5297,"children":5298,"id":5302},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[5299],{"__TSPROSE_proseElement":219,"schema":5300,"data":5301},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Разность квадратов особняком","emphasis-ealRCqFXE","paragraph-vFqUVvk41",{"__TSPROSE_proseElement":219,"schema":5305,"children":5306,"id":5316},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5307,5310],{"__TSPROSE_proseElement":219,"schema":5308,"data":5309},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Формулы, названия которых начинаются со степени (\"квадрат ...\" и \"куб ...\"), имеют похожую форму и их можно вывести естественным образом через раскрытие скобок ",{"__TSPROSE_proseElement":219,"schema":5311,"data":5312,"storageKey":5314,"id":5315},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5313},"(a+b)^2 = (a+b)(a+b) = \\ldots","$ (a+b)^2 = (a+b)(a+b) = \\ldots $","inlinerMath-E2hB8EXr8","paragraph-GM7HAKhf4",{"__TSPROSE_proseElement":219,"schema":5318,"children":5319,"id":5347},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5320,5323,5329,5332,5336,5339,5345],{"__TSPROSE_proseElement":219,"schema":5321,"data":5322},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"А вот разность квадратов ",{"__TSPROSE_proseElement":219,"schema":5324,"data":5325,"storageKey":5327,"id":5328},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5326},"a^2-b^2","$ a^2-b^2 $","inlinerMath-Z18OlR0cJ",{"__TSPROSE_proseElement":219,"schema":5330,"data":5331},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," стоит особняком. Во-первых, она раскладывается в скобки с плюсом и минусом. Во-вторых из формы ",{"__TSPROSE_proseElement":219,"schema":5333,"data":5334,"storageKey":4053,"id":5335},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":4052},"inlinerMath-T1YOXnC6X-4",{"__TSPROSE_proseElement":219,"schema":5337,"data":5338},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," нельзя естественным и явным образом получить из самой разности произведение ",{"__TSPROSE_proseElement":219,"schema":5340,"data":5341,"storageKey":5343,"id":5344},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":5342},"(a+b)(a-b)","$ (a+b)(a-b) $","inlinerMath-bZ4wF1qyF",{"__TSPROSE_proseElement":219,"schema":5346,"data":763},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"paragraph-TjiEV1CkX",{"__TSPROSE_proseElement":219,"schema":5349,"children":5350},{"name":5217,"type":222,"linkable":223,"__TSPROSE_schema":219},[5351,5363,5383,5389],{"__TSPROSE_proseElement":219,"schema":5352,"children":5353,"id":5362},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5354],{"__TSPROSE_proseElement":219,"schema":5355,"data":5356,"children":5357,"id":5361},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[5358],{"__TSPROSE_proseElement":219,"schema":5359,"data":5360},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Какая степень -- такой и коэффициент","emphasis-KSb50gTxe","paragraph-DJoU9TSeB",{"__TSPROSE_proseElement":219,"schema":5364,"children":5365,"id":5382},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5366,5369,5373,5376,5380],{"__TSPROSE_proseElement":219,"schema":5367,"data":5368},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"В квадрате\u002Fкубе суммы\u002Fразности степень (вторая или третья) встречается и как коэффициент в разложении. Для квадрата суммы\u002Fразности это ",{"__TSPROSE_proseElement":219,"schema":5370,"data":5371,"storageKey":529,"id":5372},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":528},"inlinerMath-z9Yar9waf-13",{"__TSPROSE_proseElement":219,"schema":5374,"data":5375},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},", для куба суммы\u002Fразности это ",{"__TSPROSE_proseElement":219,"schema":5377,"data":5378,"storageKey":4744,"id":5379},{"name":327,"type":238,"linkable":219,"__TSPROSE_schema":219},{"katex":4743},"inlinerMath-mqn34wvB0-1",{"__TSPROSE_proseElement":219,"schema":5381,"data":1188},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"paragraph-HH041xis9",{"__TSPROSE_proseElement":219,"schema":5384,"data":5385,"storageKey":5387,"id":5388},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5386,"freeze":223},"(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":219,"schema":5390,"children":5391,"id":5417},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5392,5395,5403,5406,5414],{"__TSPROSE_proseElement":219,"schema":5393,"data":5394},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"А еще коэффициент легко вспоминть по геометрическому выводу формул. Для \"квадрата ...\" мы составляем квадрат и в процессе появляются ",{"__TSPROSE_proseElement":219,"schema":5396,"data":5397,"children":5398,"id":5402},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[5399],{"__TSPROSE_proseElement":219,"schema":5400,"data":5401},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"два","emphasis-lF0cBJQwp",{"__TSPROSE_proseElement":219,"schema":5404,"data":5405},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," прямоугольника. А для \"куба ...\" мы составляем куб и в процессе появляются два вида из ",{"__TSPROSE_proseElement":219,"schema":5407,"data":5408,"children":5409,"id":5413},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[5410],{"__TSPROSE_proseElement":219,"schema":5411,"data":5412},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"трех","emphasis-dhAIDBC6v",{"__TSPROSE_proseElement":219,"schema":5415,"data":5416},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," параллелепипедов.","paragraph-rze0H5eSn","memorization-tips",{"__TSPROSE_proseElement":219,"schema":5420,"data":5421,"id":5423},{"name":227,"type":222,"linkable":219,"__TSPROSE_schema":219},{"level":172,"title":5422},"Сумма и разность высших степеней","summa-i-raznost-vysshikh-stepeney",{"__TSPROSE_proseElement":219,"schema":5425,"children":5426,"id":5430},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5427],{"__TSPROSE_proseElement":219,"schema":5428,"data":5429},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Заметили, что чем больше степень, тем сложнее и длиннее становятся формулы сокращенного умножения? Можно ли продолжать повышать степень бесконечно? Можно ли найти ультимативную разгадку \"Мечты первокурсника\"?","paragraph-esvna67X0",{"__TSPROSE_proseElement":219,"schema":5432,"data":5433,"storageKey":5435,"id":5436},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5434,"freeze":223},"(a \\pm b)^1 = a \\pm b \\\\ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 \\\\ \\text{???}","$$ (a \\pm b)^1 = a \\pm b \\\\ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 \\\\ \\text{???} $$","blockMath-nN28RRSij",{"__TSPROSE_proseElement":219,"schema":5438,"children":5439,"id":5454},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5440,5443,5451],{"__TSPROSE_proseElement":219,"schema":5441,"data":5442},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"На самом деле можно. Существует мега-крутая универсальная формула, которая автоматически выдает формулы сокращенного умножения для абсолютно любой степени! Называется она ",{"__TSPROSE_proseElement":219,"schema":5444,"data":5445,"children":5446,"id":5450},{"name":631,"type":238,"linkable":219,"__TSPROSE_schema":219},{"type":633},[5447],{"__TSPROSE_proseElement":219,"schema":5448,"data":5449},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Бином Ньютона","emphasis-tHYybhG8n",{"__TSPROSE_proseElement":219,"schema":5452,"data":5453},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219}," и выглядит вот так:","paragraph-CeFjI0pTW",{"__TSPROSE_proseElement":219,"schema":5456,"data":5457,"storageKey":5459,"id":5460},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5458,"freeze":223},"(a+b)^n = \\sum\\limits_{k=0}^{n} \\binom{n}{k} a^{n-k}b^k, \\quad \\text{где } \\binom{n}{k} = \\frac{n!}{k!(n-k)!}","$$ (a+b)^n = \\sum\\limits_{k=0}^{n} \\binom{n}{k} a^{n-k}b^k, \\quad \\text{где } \\binom{n}{k} = \\frac{n!}{k!(n-k)!} $$","blockMath-LZ441Tc09",{"__TSPROSE_proseElement":219,"schema":5462,"children":5463,"id":5467},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5464],{"__TSPROSE_proseElement":219,"schema":5465,"data":5466},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Вы скорее всего в шоке. Это нормально, формула действительно выглядит страшно, а для её вывода надо разбираться в комбинаторике, в учебнике про которую она и выводится. Причём никакой высшей математики не требуется, и получить её можно на \"школьном\" уровне знаний!","paragraph-1XamjhkAx",{"__TSPROSE_proseElement":219,"schema":5469,"children":5470,"id":5474},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5471],{"__TSPROSE_proseElement":219,"schema":5472,"data":5473},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Так что ответ на вопрос про бесконечное количество формул сокращенного умножения утвердительный. Да, можно бесконечно повышать степень и получать все новые и новые формулы:","paragraph-QixLIs3Pk",{"__TSPROSE_proseElement":219,"schema":5476,"data":5477,"storageKey":5479,"id":5480},{"name":267,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":5478,"freeze":223},"(a \\pm b)^4 = a^4 \\pm 4a^3b + 6a^2b^2 \\pm 4ab^3 + b^4 \\\\ (a \\pm b)^5 = a^5 \\pm 5a^4b + 10a^3b^2 \\pm 10a^2b^3 + 5ab^4 \\pm b^5 \\\\ \\ldots","$$ (a \\pm b)^4 = a^4 \\pm 4a^3b + 6a^2b^2 \\pm 4ab^3 + b^4 \\\\ (a \\pm b)^5 = a^5 \\pm 5a^4b + 10a^3b^2 \\pm 10a^2b^3 + 5ab^4 \\pm b^5 \\\\ \\ldots $$","blockMath-USBHSmZHI",{"__TSPROSE_proseElement":219,"schema":5482,"children":5483,"id":5487},{"name":233,"type":222,"linkable":219,"__TSPROSE_schema":219},[5484],{"__TSPROSE_proseElement":219,"schema":5485,"data":5486},{"name":237,"type":238,"linkable":223,"__TSPROSE_schema":219},"Бином Ньютона вообще очень часто встречается в самых разных разделах математики, и в базовой, а в особенности в высшей. Подобно тому, как ФСУ позволяют перегонять суммы в произведения скобок и наоборот для небольших степеней, бином Ньютона позволяет проворачивать это с выражениями любой сложности.","paragraph-pJY8XdQAR",{"$$ (x+5)^2 - (x-5)(x+5) - 10(x+5) $$":5489,"$$ 0 $$":5489,"$$ (x+5)^2 = (x+5)(x+5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25 $$":5489,"$$ (x-5)(x+5) = x^2 + \\cancel{5x} - \\cancel{5x} - 25 = x^2 - 25 $$":5489,"$ +5x $":5489,"$ -5x $":5489,"$$ 10(x+5) = 10x + 50 $$":5489,"$$ (x^2 + 10x + 25) - (x^2 - 25) - (10x + 50) = \\cancel{x^2} + \\cancel{10x} + 25 - \\cancel{x^2} + 25 - \\cancel{10x} - 50 = \\boxed{0} $$":5489,"$$ \\frac{(a+b)^2 - (a-b)^2}{4ab} $$":5489,"$$ 1 $$":5489,"$$ (a+b)^2 = (a + b)(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2 \\\\ (a-b)^2 = (a - b)(a-b) = a^2 - ab - ba + b^2 = a^2 - 2ab + b^2 $$":5489,"$$ (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = \\cancel{a^2} + 2ab + \\cancel{b^2} - \\cancel{a^2} + 2ab - \\cancel{b^2} = 4ab $$":5489,"$ 4ab $":5489,"$$ \\frac{\\cancel{4ab}}{\\cancel{4ab}} = \\boxed{1} $$":5489,"$$ \\frac{(m+n)^2 - m^2 - n^2}{mn} \\cdot \\frac{(m-n)^2 - m^2 - n^2}{mn} $$":5489,"$$ -4 $$":5489,"$$ (m+n)^2 - m^2 - n^2 = \\cancel{m^2} + 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = 2mn $$":5489,"$$ (m-n)^2 - m^2 - n^2 = \\cancel{m^2} - 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = -2mn $$":5489,"$$ \\frac{2\\cancel{mn}}{\\cancel{mn}} \\cdot \\frac{-2\\cancel{mn}}{\\cancel{mn}} = 2 \\cdot (-2) = \\boxed{-4} $$":5489,"$ 2 $":5489,"$ 90% $":5489,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fa-b-squared-meme.svg":5490,"$ (a+b)^n = a^n + b^n $":5489,"\u003Clink:external>\u002Fhttps:\u002F\u002Fw.wiki\u002FPjb":5494,"$$ \\red{(1+2)^2 = 1^2 + 2^2 = 5} \\\\ \\boxed{\\green{(1 + 2)^2 = 3^2 = 9}} >>{big} \\red{(2+3)^3 = 2^3 + 3^3 = 35} \\\\ \\boxed{\\green{(2 + 3)^3 = 5^3 = 125}} $$":5489,"$ (a+b)^2 $":5489,"$ a + b $":5489,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Ffountain-method.svg":5499,"$$ (a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \\boxed{a^2 + 2ab + b^2} $$":5489,"$ 2ab $":5489,"$$ a^2 + 2ab + b^2 = a^2 + ab + ab + b^2 = a(a + b) + b(a + b) = (a + b)(a + b) = \\boxed{(a + b)^2} $$":5489,"$ a $":5489,"$ b $":5489,"$ a^2 $":5489,"$ ab $":5489,"$ b^2 $":5489,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-sum-schema.svg":5503,"$$ (a + b)^2 = a^2 + 2ab + b^2 $$":5489,"$$ (m+5)^2 $$":5489,"$$ m^2 + 10m + 25 $$":5489,"$ m $":5489,"$ 5 $":5489,"$$ m^2 >>{big} 2 \\cdot m \\cdot 5 = 10m >>{big} 5^2 = 25 $$":5489,"$$ (m+5)^2 = m^2 + 10m + 25 $$":5489,"$$ \\left( 2 + \\frac{1}{8}x \\right)^2 $$":5489,"$$ 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$":5489,"$ \\frac{1}{8}x $":5489,"$$ 2^2 = 4 >>{big} \\cancel{2} \\cdot \\cancel{2} \\cdot \\frac{1}{\\cancel{8}_{\\small\\cancel{4}_{\\small 2}}}x = \\frac{1}{2}x >>{big} \\left( \\frac{1}{8}x \\right)^2 = \\left(\\frac{1}{8}\\right)^2 x^2 = \\frac{1}{64}x^2 $$":5489,"$$ \\left( 2 + \\frac{1}{8}x \\right)^2 = 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$":5489,"$$ (-7a - 3b)^2 $$":5489,"$ 1 $":5489,"$$ (-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2 $$":5489,"$$ 49a^2 + 42ab + 9b^2 $$":5489,"$ -7a - 3b $":5489,"$ -7a $":5489,"$ 3b $":5489,"$ -3b $":5489,"$$ (-7a)^2 = (-7)^2 a^2 = 49a^2 >>{big} 2 \\cdot (-7a) \\cdot (-3b) = 42ab >>{big} (-3b)^2 = (-3)^2 b^2 = 9b^2 $$":5489,"$$ (-7a - 3b)^2 = 49a^2 + 42ab + 9b^2 $$":5489,"$$ -7a - 3b = -(7a + 3b) $$":5489,"$$ (-(7a + 3b))^2 = (-1)^2 \\cdot (7a + 3b)^2 = (7a + 3b)^2 = 49a^2 + 42ab + 9b^2 $$":5489,"$$ 49 + 14x + x^2 $$":5489,"$$ (7 + x)^2 $$":5489,"$ 49 = 7^2 $":5489,"$ x^2 = x^2 $":5489,"$ a = 7 $":5489,"$ b = x $":5489,"$ 2ab = 2 \\cdot 7 \\cdot x = 14x $":5489,"$$ 49 + 14x + x^2 = \\underset{a^2}{7^2} + 2 \\cdot \\underset{a}{7} \\cdot \\underset{b}{x} + \\underset{b^2}{x^2} = (7 + x)^2 $$":5489,"$ 14x $":5489,"$ 7x $":5489,"$ 49 $":5489,"$ x^2 $":5489,"$ 7x = 7 \\cdot x $":5489,"$$ 1 + 8y + 16y^2 $$":5489,"$ 1^2 = 1 $":5489,"$$ (1 + 4y)^2 $$":5489,"$ 1 = 1^2 $":5489,"$ 16y^2 = (4y)^2 $":5489,"$ a = 1 $":5489,"$ b = 4y $":5489,"$ 2ab = 2 \\cdot 1 \\cdot 4y = 8y $":5489,"$$ 1 + 8y + 16y^2 = \\underset{a^2}{1^2} + 2 \\cdot \\underset{a}{1} \\cdot \\underset{b}{4y} + \\underset{b^2}{(4y)^2} = (1 + 4y)^2 $$":5489,"$ 8y $":5489,"$ 4y $":5489,"$ 16y^2 $":5489,"$ 1 \\cdot 4y $":5489,"$ 1^2 = 1 = a $":5489,"$ (4y)^2 = 16y^2 = b^2 $":5489,"$$ \\frac{1}{4}k^2 + k + 1 $$":5489,"$$ \\left(\\frac{k}{2} + 1 \\right)^2 $$":5489,"$ \\frac{1}{4}k^2 = \\left(\\frac{k}{2}\\right)^2 $":5489,"$ a = \\frac{k}{2} $":5489,"$ b = 1 $":5489,"$ 2ab = 2 \\cdot \\frac{k}{2} \\cdot 1 = k $":5489,"$$ \\frac{1}{4}k^2 + k + 1 = \\underset{a^2}{\\left( \\frac{k}{2} \\right)^2} + 2 \\cdot \\underset{a}{\\frac{k}{2}} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = \\left(\\frac{k}{2} + 1 \\right)^2 $$":5489,"$ k $":5489,"$ \\frac{k}{2} $":5489,"$ \\frac{1}{4}k^2 $":5489,"$ \\frac{k}{2} \\cdot 1 $":5489,"$ \\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2 $":5489,"$ 1^2 = 1 = b^2 $":5489,"$$ 16t^2 + 36m^2 + 48tm $$":5489,"$$ (4t + 6m)^2 $$":5489,"$$ 16t^2 + 48tm + 36m^2 $$":5489,"$ 16t^2 = (4t)^2 $":5489,"$ 36m^2 = (6m)^2 $":5489,"$ a = 4t $":5489,"$ b = 6m $":5489,"$ 2ab = 2 \\cdot 4t \\cdot 6m = 48tm $":5489,"$$ 16t^2 + 36m^2 + 48tm = \\underset{a^2}{(4t)^2} + 2 \\cdot \\underset{a}{4t} \\cdot \\underset{b}{6m} + \\underset{b^2}{(6m)^2} = (4t + 6m)^2 $$":5489,"$ 48tm $":5489,"$ 24tm $":5489,"$ 16t^2 $":5489,"$ 36m^2 $":5489,"$ 24tm = 4t \\cdot 6m $":5489,"$ (4t)^2 = 16t^2 = a^2 $":5489,"$ (6m)^2 = 36m^2 = b^2 $":5489,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fancient-formula.webp":5507,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas":5509,"$ (a-b)^2 $":5489,"$ a-b $":5489,"$$ (a - b)^2 = (a - b)(a - b) = a^2 - ab - ba + b^2 = \\boxed{a^2 - 2ab + b^2} \\\\ a^2 - 2ab + b^2 = a^2 - ab - ab + b^2 = a(a - b) - b(a - b) = (a - b)(a - b) = \\boxed{(a - b)^2} $$":5489,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-diff-schema.svg":5517,"$$ (a - b)^2 = a^2 - 2ab + b^2 $$":5489,"$ 3-2 $":5489,"$ 3+(-2) $":5489,"$$ - a - b + c - d = (-a) + (-b) + c + (-d) $$":5489,"$ -2 $":5489,"$$ (6-c)^2 $$":5489,"$$ 36 - 12c + c^2 $$":5489,"$ 6 $":5489,"$ c $":5489,"$$ 6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2 $$":5489,"$$ (6-c)^2 = 36 - 12c + c^2 $$":5489,"$ -b $":5489,"$$ (6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2} $$":5489,"$$ 9x^2 - 6x + 1 $$":5489,"$$ (3x - 1)^2 $$":5489,"$ 9x^2 = (3x)^2 $":5489,"$ a = 3x $":5489,"$ -2ab = -2 \\cdot 3x \\cdot 1 = -6x $":5489,"$$ 9x^2 - 6x + 1 = \\underset{a^2}{(3x)^2} - 2 \\cdot \\underset{a}{3x} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = (3x - 1)^2 $$":5489,"$ -6x $":5489,"$ 3x $":5489,"$ 9x^2 $":5489,"$ 3x = 3x \\cdot 1 $":5489,"$ 9x^2 = (3x)^2 = a^2 $":5489,"$ 1 = 1^2 = b^2 $":5489,"$$ (-7 + 2a)^2 $$":5489,"$$ 4a^2 - 28a + 49 $$":5489,"$$ (-7 + 2a)^2 = (2a - 7)^2 $$":5489,"$ 2a $":5489,"$ 7 $":5489,"$$ (2a)^2 = 4a^2 >>{big} -2 \\cdot 2a \\cdot 7 = -28a >>{big} 7^2 = 49 $$":5489,"$$ (-7 + 2a)^2 = 4a^2 - 28a + 49 $$":5489,"$$ - 12k + 4k^2 + 9 $$":5489,"$$ (2k - 3)^2 $$":5489,"$$ - 12k + 4k^2 + 9 = 4k^2 - 12k + 9 $$":5489,"$ 4k^2 = (2k)^2 $":5489,"$ 9 = 3^2 $":5489,"$ a = 2k $":5489,"$ b = 3 $":5489,"$ -2ab = -2 \\cdot 2k \\cdot 3 = -12k $":5489,"$$ 4k^2 - 12k + 9 = \\underset{a^2}{(2k)^2} - 2 \\cdot \\underset{a}{2k} \\cdot \\underset{b}{3} + \\underset{b^2}{3^2} = (2k - 3)^2 $$":5489,"$ -12k $":5489,"$ 6k $":5489,"$ 4k^2 $":5489,"$ 9 $":5489,"$ 6k = 2k \\cdot 3 $":5489,"$ (2k)^2 = 4k^2 = a^2 $":5489,"$ 3^2 = 9 = b^2 $":5489,"$$ \\left( 5y - \\frac{3}{4}x \\right)^2 $$":5489,"$$ 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$":5489,"$ 5y $":5489,"$ \\frac{3}{4}x $":5489,"$$ (5y)^2 = 5^2y^2 = 25y^2 >>{big} -\\cancel{2} \\cdot 5y \\cdot \\frac{3}{\\cancel{4}_{\\small 2}}x = -\\frac{15}{2}xy >>{big} \\left( \\frac{3}{4}x \\right)^2 = \\left(\\frac{3}{4}\\right)^2 x^2 = \\frac{9}{16}x^2 $$":5489,"$$ \\left( 5y - \\frac{3}{4}x \\right)^2 = 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$":5489,"$$ \\frac{1}{4}m^2 - 5m + 25 $$":5489,"$$ \\left(\\frac{m}{2} - 5\\right)^2 $$":5489,"$ \\frac{1}{4}m^2 = \\left(\\frac{m}{2}\\right)^2 $":5489,"$ 25 = 5^2 $":5489,"$ a = \\frac{m}{2} $":5489,"$ b = 5 $":5489,"$ -2ab = -2 \\cdot \\frac{m}{2} \\cdot 5 = -5m $":5489,"$$ \\frac{1}{4}m^2 - 5m + 25 = \\underset{a^2}{\\left(\\frac{1}{2}m\\right)^2} - 2 \\cdot \\underset{a}{\\frac{1}{2}m} \\cdot \\underset{b}{5} + \\underset{b^2}{5^2} = \\left(\\frac{m}{2} - 5\\right)^2 $$":5489,"$ -5m $":5489,"$ \\frac{5m}{2} $":5489,"$ \\frac{1}{4}m^2 $":5489,"$ 25 $":5489,"$ \\frac{5m}{2} = \\frac{m}{2} \\cdot 5 $":5489,"$ \\left(\\frac{m}{2}\\right)^2 = \\frac{1}{4}m^2 = a^2 $":5489,"$ 5^2 = 25 = b^2 $":5489,"$ 0 $":5489,"$ 10 $":5489,"$ 4^2 = 16 $":5489,"$ 6^2 = 36 $":5489,"$ 9^2 = 81 $":5489,"$ 20 $":5489,"$ 11^2 = 121 $":5489,"$ 15^2 = 225 $":5489,"$ 19^2 = 361 $":5489,"$ 99% $":5489,"$ > 15 $":5489,"$ 62 $":5489,"$ 60 $":5489,"$ 3844 $":5489,"$ 62 = 60 + 2 $":5489,"$ 62^2 $":5489,"$$ 62^2 = (60 + 2)^2 = 60^2 + 2 \\cdot 60 \\cdot 2 + 2^2 = 3600 + 240 + 4 = 3844 $$":5489,"$ 2 \\cdot 60 \\cdot 2 $":5489,"$ 3600 $":5489,"$ 240 $":5489,"$$ 24 \\Rightarrow 20 + 4 >>{big} 31 \\Rightarrow 30 + 1 >>{big} 44 \\Rightarrow 40 + 4 >>{big} 53 \\Rightarrow 50 + 3 $$":5489,"$ 48 $":5489,"$ 50 $":5489,"$ 2304 $":5489,"$ 48 = 50 - 2 $":5489,"$ 48^2 $":5489,"$$ 48^2 = (50 - 2)^2 = 50^2 - 2 \\cdot 50 \\cdot 2 + 2^2 = 2500 - 200 + 4 = 2304 $$":5489,"$ 2 \\cdot 50 \\cdot 2 $":5489,"$$ 27 \\Rightarrow 30 - 3 >>{big} 39 \\Rightarrow 40 - 1 >>{big} 46 \\Rightarrow 50 - 4 >>{big} 57 \\Rightarrow 60 - 3 $$":5489,"problemScript:content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fmental-squares":5521,"$ 42 $":5489,"$ 1764 $":5489,"$$ x^2 + 2x + 1 = 0 >>{big} 36 - 18t + 9t^2 = 0 >>{big} 4z^2 + 48z + 144 = 0 $$":5489,"$ 0 = 0 $":5489,"$$ (x+1)^2 = 0 >>{big} (6 - 3t)^2 = 0 >>{big} (2z + 12)^2 = 0 $$":5489,"$ x $":5489,"$ -1 $":5489,"$ t $":5489,"$ z $":5489,"$ 12 $":5489,"$ -6 $":5489,"$$ (\\underset{x}{-1} + 1)^2 = 0 >>{big} (6 - 3 \\cdot \\underset{t}{2})^2 = 0 >>{big} (2 \\cdot \\underset{z}{-6} + 12)^2 = 0 $$":5489,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square":5523,"$ a^2 - b^2 $":5489,"$ a - b $":5489,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fdiff-of-squares.svg":5527,"$$ a^2 - b^2 = (a + b)(a - b) $$":5489,"$$ (a+b)(a-b) = a \\cdot a - a \\cdot b + b \\cdot a - b \\cdot b = \\boxed{a^2 - b^2} $$":5489,"$$ a^2 - b^2 = a^2 + \\underbrace{ab - ab}_{\\text{Добавили и вычли}} - b^2 = a(a+b) - b(a+b) = \\boxed{(a+b)(a-b)} $$":5489,"$ (a - b)^2 $":5489,"$$ x^2 - 36 $$":5489,"$$ (x + 6)(x - 6) $$":5489,"$ x \\Rightarrow x^2 $":5489,"$ 6 \\Rightarrow 36 $":5489,"$$ x^2 - 36 = \\underset{a}{x^2} - \\underset{b}{6^2} = (x + 6)(x - 6) $$":5489,"$$ (3 - x)(3 + x) $$":5489,"$$ 9 - x^2 $$":5489,"$$ 3^2 = 9 >>{big} x^2 $$":5489,"$$ \\frac{4}{81}k^2 - \\frac{1}{25}b^2 $$":5489,"$$ \\left(\\frac{2k}{9} + \\frac{b}{5}\\right)\\left(\\frac{2k}{9} - \\frac{b}{5}\\right) $$":5489,"$$ \\frac{4}{81}k^2 = \\frac{2^2}{9^2}k^2 = \\left(\\frac{2k}{9}\\right)^2 >>{big} \\frac{1}{25}b^2 = \\frac{1^2}{5^2}b^2 = \\left(\\frac{b}{5}\\right)^2 $$":5489,"$ 2k\u002F9 $":5489,"$ b\u002F5 $":5489,"$$ \\frac{4}{81}k^2 - \\frac{1}{25}b^2 = \\underset{a}{\\left(\\frac{2k}{9}\\right)^2} - \\underset{b}{\\left(\\frac{b}{5}\\right)^2} = \\left(\\frac{2k}{9} + \\frac{b}{5}\\right)\\left(\\frac{2k}{9} - \\frac{b}{5}\\right) $$":5489,"$$ \\left( t + \\frac{3}{4} \\right)\\left( \\frac{3}{4} - t \\right) $$":5489,"$$ \\frac{9}{16} - t^2 $$":5489,"$$ \\left( t + \\frac{3}{4} \\right)\\left( \\frac{3}{4} - t \\right) = \\left( \\frac{3}{4} + t \\right)\\left( \\frac{3}{4} - t \\right) = \\ldots $$":5489,"$$ \\ldots = \\left( \\frac{3}{4} + t \\right)\\left( \\frac{3}{4} - t \\right) = \\left(\\frac{3}{4}\\right)^2 - t^2 = \\frac{9}{16} - t^2 $$":5489,"$$ 9x^4z^2 - 0.09y^2 $$":5489,"$$ (3x^2z + 0.3y)(3x^2z - 0.3y) $$":5489,"$$ 9x^4z^2 = 3^2(x^2)^2z^2 = (3x^2z)^2 >>{big} 0.09y^2 = \\frac{9y^2}{100} = \\frac{3^2y^2}{10^2} = \\left(\\frac{3y}{10}\\right)^2 = (0.3y)^2 $$":5489,"$ 3x^2z $":5489,"$ 0.3y $":5489,"$$ 9x^4z^2 - 0.09y^2 = \\underset{a}{(3x^2z)^2} - \\underset{b}{(0.3y)^2} = (3x^2z + 0.3y)(3x^2z - 0.3y) $$":5489,"$$ (a+b)^3 = (a+b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = \\boxed{a^3 + 3a^2b + 3ab^2 + b^3} \\\\ (a-b)^3 = (a-b)^2(a-b) = (a^2 - 2ab + b^2)(a-b) = \\boxed{a^3 - 3a^2b + 3ab^2 - b^3} $$":5489,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fpractice\u002F$cubeSumDiffFactorization":5531,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fcube-sum.mp4":5539,"\u003Clink:external>\u002Fhttps:\u002F\u002Fwww.tiktok.com\u002F@complex_math\u002Fvideo\u002F7358570064724970759":5541,"$ 3 $":5489,"$ a^2b $":5489,"$ ab^2 $":5489,"$$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$":5489,"$$ (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $$":5489,"$$ (x + 2)^3 $$":5489,"$$ x^3 + 6x^2 + 12x + 8 $$":5489,"$$ (x + 2)^3 = x^3 + 3 \\cdot x^2 \\cdot 2 + 3 \\cdot x \\cdot 2^2 + 2^3 = x^3 + 6x^2 + 12x + 8 $$":5489,"$$ \\left(\\frac{1}{3} - \\frac{2}{k}\\right)^3 $$":5489,"$$ \\frac{1}{27} - \\frac{2}{3k} + \\frac{4}{k^2} - \\frac{8}{k^3} $$":5489,"$ 1\u002F3 $":5489,"$ 2\u002Fk $":5489,"$$ \\left(\\frac{1}{3} - \\frac{2}{k}\\right)^3 = \\left(\\frac{1}{3}\\right)^3 - 3\\left(\\frac{1}{3}\\right)^2 \\cdot \\frac{2}{k} + 3 \\cdot \\frac{1}{3} \\cdot \\left(\\frac{2}{k}\\right)^2 - \\left(\\frac{2}{k}\\right)^3 = \\\\ \\frac{1}{27} - \\cancel{3} \\cdot \\frac{1}{\\cancel{9}_{\\small 3}} \\cdot \\frac{2}{k} + \\cancel{3} \\cdot \\frac{1}{\\cancel{3}_{\\small 1}} \\cdot \\frac{4}{k^2} - \\frac{8}{k^3} = \\\\ \\frac{1}{27} - \\frac{2}{3k} + \\frac{4}{k^2} - \\frac{8}{k^3} $$":5489,"$$ m^3 + 6m^2n + 12mn^2 + 8n^3 $$":5489,"$$ (m + 2n)^3 $$":5489,"$ a^3 = m^3 $":5489,"$ a = m $":5489,"$ b^3 = 8n^3 = (2n)^3 $":5489,"$ b = 2n $":5489,"$$ \\underbrace{m^3}_{a^3} + 3 \\cdot \\underbrace{m^2}_{a^2} \\cdot \\underbrace{2n}_{b} + 3 \\cdot \\underbrace{m}_{a} \\cdot \\underbrace{(2n)^2}_{b^2} + \\underbrace{(2n)^3}_{b^3} = (m + 2n)^3 $$":5489,"$$ \\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} $$":5489,"$$ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3 $$":5489,"$ a^3 = x^3\u002F8 = (x\u002F2)^3 $":5489,"$ a = x\u002F2 $":5489,"$ b^3 = y^3\u002F27 = (y\u002F3)^3 $":5489,"$ b = y\u002F3 $":5489,"$$ \\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} = \\\\ \\underbrace{\\frac{x^3}{8}}_{a^3} - 3 \\cdot \\underbrace{\\frac{x^2}{4}}_{a^2} \\cdot \\underbrace{\\frac{y}{3}}_{b} + 3 \\cdot \\underbrace{\\frac{x}{2}}_{a} \\cdot \\underbrace{\\frac{y^2}{9}}_{b^2} - \\underbrace{\\frac{y^3}{27}}_{b^3} = \\\\ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3 $$":5489,"$ (a + b)^2 $":5489,"$ 4 $":5489,"$$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $$":5489,"$ (a+b)^2 = (a+b)(a+b) = \\ldots $":5489,"$ a^2-b^2 $":5489,"$ (a+b)(a-b) $":5489,"$$ (a \\pm b)^{\\normalsize\\brand{2}} = a^2 \\pm \\brand{2}ab + b^2 \\\\ (a \\pm b)^{\\normalsize\\brand{3}} = a^3 \\pm \\brand{3}a^2b + \\brand{3}ab^2 \\pm b^3 $$":5489,"$$ (a \\pm b)^1 = a \\pm b \\\\ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 \\\\ \\text{???} $$":5489,"$$ (a+b)^n = \\sum\\limits_{k=0}^{n} \\binom{n}{k} a^{n-k}b^k, \\quad \\text{где } \\binom{n}{k} = \\frac{n!}{k!(n-k)!} $$":5489,"$$ (a \\pm b)^4 = a^4 \\pm 4a^3b + 6a^2b^2 \\pm 4ab^3 + b^4 \\\\ (a \\pm b)^5 = a^5 \\pm 5a^4b + 10a^3b^2 \\pm 10a^2b^3 + 5ab^4 \\pm b^5 \\\\ \\ldots $$":5489},null,{"resolvedSrc":5491,"width":5492,"height":5493},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fa-b-squared-meme.svg",1079,1457,{"type":5495,"resolvedHref":5496,"previewRequest":5497},"external","https:\u002F\u002Fw.wiki\u002FPjb",{"type":5498,"href":5496},"direct-link",{"resolvedSrc":5500,"width":5501,"height":5502},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Ffountain-method.svg",942,676,{"resolvedSrc":5504,"width":5505,"height":5506},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-sum-schema.svg",4182,1355,{"resolvedIconSrc":5508,"videoIcon":223},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fancient-formula.webp",{"type":5510,"content":5511,"resolvedHref":5513,"previewRequest":5514},"contentItem",{"contentType":180,"contentTitle":5512,"topicPart":214},"Теорема Виета","\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas\u002F",{"type":5515,"contentType":180,"topicPart":214,"fullId":5516},"content-page","foundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas",{"resolvedSrc":5518,"width":5519,"height":5520},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-diff-schema.svg",4363,1418,{"resolvedScriptSrc":5522},"\u002Fapi\u002FproblemScript\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fmental-squares.js",{"type":5510,"content":5524,"resolvedHref":182,"previewRequest":5525},{"contentType":180,"contentTitle":181,"topicPart":214},{"type":5515,"contentType":180,"topicPart":214,"fullId":5526},"foundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square",{"resolvedSrc":5528,"width":5529,"height":5530},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fdiff-of-squares.svg",5848,1384,{"type":5532,"schemaName":5533,"elementTitle":5534,"content":5535,"resolvedHref":5536,"previewRequest":5537},"unique","problem","Факторизация куба суммы и разности",{"contentType":180,"contentTitle":159,"topicPart":217},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum-diff-factorization",{"type":5532,"contentFullId":157,"contentType":180,"topicPart":217,"uniqueName":5538},"cubeSumDiffFactorization",{"resolvedSrc":5540},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fcube-sum.mp4",{"type":5495,"resolvedHref":5542,"previewRequest":5543},"https:\u002F\u002Fwww.tiktok.com\u002F@complex_math\u002Fvideo\u002F7358570064724970759",{"type":5498,"href":5542},[5545,5550,5555,5559,5563,5568,5574,5579,5585,5591,5597,5605,5610,5615],{"link":5546,"schemaName":227,"title":229,"seo":5547,"description":5549},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=chto-takoe-fsu",{"title":5548},"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":5551,"schemaName":537,"title":539,"key":5552,"seo":5553,"description":5554},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=what-are-special-products",{"title":229},{},"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":187,"schemaName":188,"title":557,"key":5556,"seo":5557,"description":5558},{},{},"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":192,"schemaName":188,"title":2116,"key":5560,"seo":5561,"description":5562},{},{},"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":5564,"schemaName":227,"title":3245,"seo":5565,"description":5567},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=primenenie-kvadrata-summy-ili-raznosti",{"title":5566},"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":5569,"schemaName":243,"title":3390,"key":5570,"seo":5571,"description":5573},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=fast-square",{},{"title":5572},"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":5575,"schemaName":188,"title":4041,"key":5576,"seo":5577,"description":5578},"\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":5580,"schemaName":188,"title":5581,"key":5582,"seo":5583,"description":5584},"\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":5586,"schemaName":188,"title":5587,"key":5588,"seo":5589,"description":5590},"\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":5592,"schemaName":5211,"title":5177,"key":5593,"seo":5594,"description":5596},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=memorization-tips",{},{"title":5595},"Как запомнить формулы сокращенного умножения?","Набор советов и приемов, которые помогут наизусть запомнить формулы сокращенного умножения (ФСУ) и не путаться в них.",{"link":5598,"schemaName":5599,"title":5600,"key":5601,"seo":5603,"description":5604},"\u002Fsummary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=special-products-table","table","Таблица формул сокращенного умножения",{"title":5602},"Таблица ФСУ",{},"Единая таблица со всеми формулами сокращенного умножения.",{"link":5606,"schemaName":243,"title":5607,"seo":5608,"description":5609},"\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":5611,"schemaName":243,"title":5612,"seo":5613,"description":5614},"\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":5616,"schemaName":243,"title":5617,"seo":5618,"description":5619},"\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.",[5621,5623,5625,5627,5633,5635,5637,5639],{"type":227,"level":172,"title":229,"elementId":230,"children":5622},[],{"type":227,"level":172,"title":557,"elementId":558,"children":5624},[],{"type":227,"level":172,"title":2116,"elementId":2117,"children":5626},[],{"type":227,"level":172,"title":3245,"elementId":3246,"children":5628},[5629,5631],{"type":227,"level":174,"title":3257,"elementId":3258,"children":5630},[],{"type":227,"level":174,"title":3881,"elementId":3882,"children":5632},[],{"type":227,"level":172,"title":4041,"elementId":4042,"children":5634},[],{"type":227,"level":172,"title":4671,"elementId":4672,"children":5636},[],{"type":227,"level":172,"title":5177,"elementId":5178,"children":5638},[],{"type":227,"level":172,"title":5422,"elementId":5423,"children":5640},[],{"contributorsCount":5642,"sponsorsCount":173},7,{"shortBookId":5,"frontNav":5644},{"type":14,"shortId":5,"title":15,"flags":5645,"link":17,"children":5646},{},[5647,5656],{"type":5648,"separator":219,"shortId":5649,"title":165,"flags":5650,"link":166,"children":5651},"group","foundations\u002Fpolynomials",{},[5652],{"type":180,"shortId":157,"title":159,"flags":5653,"link":5654,"parts":5655},{},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F",[214,216,217],{"type":5648,"separator":219,"shortId":5657,"title":5658,"flags":5659,"link":5660,"children":5661},"foundations\u002Fequations","Уравнения",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002F",[5662,5668,5673],{"type":180,"shortId":5663,"title":5664,"flags":5665,"link":5666,"parts":5667},"foundations\u002Fequations\u002Felementary","Элементарные уравнения",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Felementary\u002F",[214,216,217],{"type":8,"shortId":5669,"title":5670,"flags":5671,"link":5672},"foundations\u002Fequations\u002Fzero-product-property","Нулевые множители",{},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fzero-product-property\u002F",{"type":5648,"separator":223,"shortId":5674,"title":5675,"flags":5676,"link":5677,"children":5678},"foundations\u002Fequations\u002Fquadratic","Квадратные уравнения",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002Fquadratic\u002F",[5679,5685,5691,5695,5701,5707,5712,5715,5721],{"type":180,"shortId":5680,"title":5681,"flags":5682,"link":5683,"parts":5684},"foundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it","Что это такое?",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it\u002F",[214,216,217],{"type":180,"shortId":5686,"title":5687,"flags":5688,"link":5689,"parts":5690},"foundations\u002Fequations\u002Fquadratic\u002Fincomplete","Неполная форма",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fincomplete\u002F",[214,216,217],{"type":180,"shortId":5526,"title":5692,"flags":5693,"link":182,"parts":5694},"Полный квадрат",{},[214,216,217],{"type":180,"shortId":5696,"title":5697,"flags":5698,"link":5699,"parts":5700},"foundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula","Формула корней",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002F",[214,216,217],{"type":180,"shortId":5702,"title":5703,"flags":5704,"link":5705,"parts":5706},"foundations\u002Fequations\u002Fquadratic\u002Ffactoring","Разложение на множители",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002F",[214,216,217],{"type":8,"shortId":5708,"title":5709,"flags":5710,"link":5711},"foundations\u002Fequations\u002Fquadratic\u002Freal-world","В реальной жизни",{"secondary":219},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F",{"type":180,"shortId":5516,"title":5512,"flags":5713,"link":5513,"parts":5714},{"secondary":219},[214,216,217],{"type":180,"shortId":5716,"title":5717,"flags":5718,"link":5719,"parts":5720},"foundations\u002Fequations\u002Fquadratic\u002Fmental-solving","Решение в уме",{"secondary":219,"advanced":219},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving\u002F",[214,216],{"type":180,"shortId":5722,"title":5723,"flags":5724,"link":5725,"parts":5726},"foundations\u002Fequations\u002Fquadratic\u002Fformulas","Общие формулы",{"secondary":219},"\u002Fsummary\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fformulas\u002F",[216,217],{"cameo":5728,"sponsor":5739},[5729,5730,5731,5732,5733,5734,5735,5736,5737,5738],"cameo-black-hole","cameo-confucius","cameo-dunno","cameo-einstein","cameo-euclid","cameo-gagarin","cameo-gauss","cameo-gorinich","cameo-newton","cameo-t800",[5740,5741],"sponsor-green-hedgehog","sponsor-reverse-shift",1777651272383]