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В формуле ",{"__TSPROSE_proseElement":219,"schema":3847,"data":3848,"storageKey":3684,"id":3849},{"name":279,"type":249,"linkable":219,"__TSPROSE_schema":219},{"label":3683},"dependencyInliner-oYxX6S3Zs-1",{"__TSPROSE_proseElement":219,"schema":3851,"data":3852},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," среднее слагаемое равно ",{"__TSPROSE_proseElement":219,"schema":3854,"data":3855,"storageKey":3857,"id":3858},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":3856},"2ab","$ 2ab $","inlinerMath-McBVEYBhA",{"__TSPROSE_proseElement":219,"schema":3860,"data":3861},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},", поэтому наша задача сейчас привести ",{"__TSPROSE_proseElement":219,"schema":3863,"data":3864,"storageKey":3866,"id":3867},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":3865},"40a^3b^4","$ 40a^3b^4 $","inlinerMath-bhAWJNQgL",{"__TSPROSE_proseElement":219,"schema":3869,"data":3870},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," к виду ",{"__TSPROSE_proseElement":219,"schema":3872,"data":3873,"storageKey":3857,"id":3874},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":3856},"inlinerMath-McBVEYBhA-1",{"__TSPROSE_proseElement":219,"schema":3876,"data":3877},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," чтобы понять, чему равно ",{"__TSPROSE_proseElement":219,"schema":3879,"data":3880,"storageKey":3882,"id":3883},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":3881},"b","$ b $","inlinerMath-LTS8nNQCZ",{"__TSPROSE_proseElement":219,"schema":3885,"data":3886},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},":","paragraph-EslrOeDJW",{"__TSPROSE_proseElement":219,"schema":3889,"data":3890,"storageKey":3892,"id":3893},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":3891,"freeze":223},"40a^3b^4 = 2 \\cdot 20a^3b^4 = 2 \\cdot \\underset{a}{(4a^3)} \\cdot \\underset{b}{(5b^4)}","$$ 40a^3b^4 = 2 \\cdot 20a^3b^4 = 2 \\cdot \\underset{a}{(4a^3)} \\cdot \\underset{b}{(5b^4)} $$","blockMath-3jkyrhaGD",{"__TSPROSE_proseElement":219,"schema":3895,"children":3896,"id":3916},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[3897,3900,3904,3907,3913],{"__TSPROSE_proseElement":219,"schema":3898,"data":3899},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Итак, роль ",{"__TSPROSE_proseElement":219,"schema":3901,"data":3902,"storageKey":3882,"id":3903},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":3881},"inlinerMath-LTS8nNQCZ-1",{"__TSPROSE_proseElement":219,"schema":3905,"data":3906},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," в формуле квадрата суммы выполняет ",{"__TSPROSE_proseElement":219,"schema":3908,"data":3909,"storageKey":3911,"id":3912},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":3910},"5b^4","$ 5b^4 $","inlinerMath-OO73kGvNA",{"__TSPROSE_proseElement":219,"schema":3914,"data":3915},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},". 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Применяем формулу разности квадратов:","paragraph-46JlkETTM",{"__TSPROSE_proseElement":219,"schema":8260,"data":8261,"storageKey":8263,"id":8264},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8262,"freeze":223},"2k + 1 = \\left[ (k+1) + k \\right] \\cdot \\left[ (k+1) - k \\right] = (k+1)^2 - k^2 \\\\ 2k + 1 = (k+1)^2 - k^2","$$ 2k + 1 = \\left[ (k+1) + k \\right] \\cdot \\left[ (k+1) - k \\right] = (k+1)^2 - k^2 \\\\ 2k + 1 = (k+1)^2 - k^2 $$","blockMath-2yB364xXk",{"__TSPROSE_proseElement":219,"schema":8266,"children":8267,"id":8272},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8268],{"__TSPROSE_proseElement":219,"schema":8269,"data":8270,"storageKey":4857,"id":8271},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":4856},"inlinerMath-77K6blkcr-4","paragraph-ea37qO3lg-4",{"__TSPROSE_proseElement":219,"schema":8274},{"name":8275,"type":222,"linkable":223,"__TSPROSE_schema":219},"hr",{"__TSPROSE_proseElement":219,"schema":8277,"children":8278,"id":8321},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8279,8282,8286,8288,8292,8294,8298,8301,8305,8308,8312,8315,8319],{"__TSPROSE_proseElement":219,"schema":8280,"data":8281},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Теперь запишем разность квадратов для чисел ",{"__TSPROSE_proseElement":219,"schema":8283,"data":8284,"storageKey":7115,"id":8285},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":7114},"inlinerMath-sYh3uEvdQ-3",{"__TSPROSE_proseElement":219,"schema":8287,"data":8045},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":8289,"data":8290,"storageKey":8050,"id":8291},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8049},"inlinerMath-eEHhsOp1e-1",{"__TSPROSE_proseElement":219,"schema":8293,"data":286},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},{"__TSPROSE_proseElement":219,"schema":8295,"data":8296,"storageKey":8058,"id":8297},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8057},"inlinerMath-b1yiDnYmR-1",{"__TSPROSE_proseElement":219,"schema":8299,"data":8300},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},". Для этого запишем все эти числа в виде ",{"__TSPROSE_proseElement":219,"schema":8302,"data":8303,"storageKey":8111,"id":8304},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8110},"inlinerMath-yWT1AujDY-3",{"__TSPROSE_proseElement":219,"schema":8306,"data":8307},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},", ведь нам нужно найти ",{"__TSPROSE_proseElement":219,"schema":8309,"data":8310,"storageKey":8146,"id":8311},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8145},"inlinerMath-KaVbgkHhg-2",{"__TSPROSE_proseElement":219,"schema":8313,"data":8314},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," для каждого числа. Для этого вычитаем из каждого единицу и делим на ",{"__TSPROSE_proseElement":219,"schema":8316,"data":8317,"storageKey":6800,"id":8318},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":6758},"inlinerMath-z9Yar9waf-3",{"__TSPROSE_proseElement":219,"schema":8320,"data":3886},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"paragraph-82rPIEFDU",{"__TSPROSE_proseElement":219,"schema":8323,"data":8324,"storageKey":8326,"id":8327},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8325,"freeze":223},"7 = 2 \\cdot 3 + 1 \\Rightarrow 3 >>{big} 111 = 2 \\cdot 55 + 1 \\Rightarrow 55 >>{big} 507 = 2 \\cdot 253 + 1 \\Rightarrow 253","$$ 7 = 2 \\cdot 3 + 1 \\Rightarrow 3 >>{big} 111 = 2 \\cdot 55 + 1 \\Rightarrow 55 >>{big} 507 = 2 \\cdot 253 + 1 \\Rightarrow 253 $$","blockMath-blIsrvaI5",{"__TSPROSE_proseElement":219,"schema":8329,"children":8330,"id":8341},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8331,8334,8338],{"__TSPROSE_proseElement":219,"schema":8332,"data":8333},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Используем найденные ",{"__TSPROSE_proseElement":219,"schema":8335,"data":8336,"storageKey":8146,"id":8337},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8145},"inlinerMath-KaVbgkHhg-3",{"__TSPROSE_proseElement":219,"schema":8339,"data":8340},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," для записи разности квадратов:","paragraph-AUTvSqDDc",{"__TSPROSE_proseElement":219,"schema":8343,"data":8344,"storageKey":8346,"id":8347},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8345,"freeze":223},"7 = \\boxed{4^2 - 3^2} >>{big} 111 = \\boxed{56^2 - 55^2} >>{big} 507 = \\boxed{254^2 - 253^2}","$$ 7 = \\boxed{4^2 - 3^2} >>{big} 111 = \\boxed{56^2 - 55^2} >>{big} 507 = \\boxed{254^2 - 253^2} $$","blockMath-UuLlIqHHc","lyuboe-nechetnoe-eto-raznost-kvadratov",{"__TSPROSE_proseElement":219,"schema":8350,"data":8351,"children":8355,"id":8587},{"name":4655,"type":222,"linkable":219,"__TSPROSE_schema":219},{"info":8352},{"title":8353,"level":8020,"attributes":8354},"Факторизация куба суммы и разности",[],[8356,8372,8382,8411,8460],{"__TSPROSE_proseElement":219,"schema":8357,"children":8358},{"name":240,"type":222,"linkable":223,"__TSPROSE_schema":219},[8359,8366],{"__TSPROSE_proseElement":219,"schema":8360,"children":8361,"id":8365},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8362],{"__TSPROSE_proseElement":219,"schema":8363,"data":8364},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Получите из развернутых выражений куба суммы и разности их запись через скобки в кубе:","paragraph-xz4OlX6SR",{"__TSPROSE_proseElement":219,"schema":8367,"data":8368,"storageKey":8370,"id":8371},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8369,"freeze":223},"a^3 + 3a^2b + 3ab^2 + b^3 \\overset{\\text{?}}{\\implies} (a+b)^3 \\\\ a^3 - 3a^2b + 3ab^2 - b^3 \\overset{\\text{?}}{\\implies} (a-b)^3","$$ a^3 + 3a^2b + 3ab^2 + b^3 \\overset{\\text{?}}{\\implies} (a+b)^3 \\\\ a^3 - 3a^2b + 3ab^2 - b^3 \\overset{\\text{?}}{\\implies} (a-b)^3 $$","blockMath-7fiCltIQJ",{"__TSPROSE_proseElement":219,"schema":8373,"children":8374},{"name":269,"type":222,"linkable":223,"__TSPROSE_schema":219},[8375],{"__TSPROSE_proseElement":219,"schema":8376,"children":8377,"id":8381},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8378],{"__TSPROSE_proseElement":219,"schema":8379,"data":8380},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Для решения вам потребуется и выносить общие множители из части слагаемых, разбивать полученные выражения на более мелкие части, вновь раскрывать скобки и опять выносить общие множители.","paragraph-sMjZoqTVs",{"__TSPROSE_proseElement":219,"schema":8383,"children":8384},{"name":269,"type":222,"linkable":223,"__TSPROSE_schema":219},[8385,8405],{"__TSPROSE_proseElement":219,"schema":8386,"children":8387,"id":8404},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8388,8391,8395,8398,8402],{"__TSPROSE_proseElement":219,"schema":8389,"data":8390},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Из первых двух слагаемых вынесите ",{"__TSPROSE_proseElement":219,"schema":8392,"data":8393,"storageKey":3776,"id":8394},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":3775},"inlinerMath-eiXlIGTfO-4",{"__TSPROSE_proseElement":219,"schema":8396,"data":8397},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},", из последних двух ",{"__TSPROSE_proseElement":219,"schema":8399,"data":8400,"storageKey":3693,"id":8401},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":3692},"inlinerMath-tBkcHvoEV-2",{"__TSPROSE_proseElement":219,"schema":8403,"data":295},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"paragraph-6C3Z2YPJg",{"__TSPROSE_proseElement":219,"schema":8406,"data":8407,"storageKey":8409,"id":8410},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8408,"freeze":223},"a^3 + 3a^2b + 3ab^2 + b^3 = a^2(a+3b) + b^2(3a+b) \\\\ a^3 - 3a^2b + 3ab^2 - b^3 = a^2(a-3b) + b^2(3a-b)","$$ a^3 + 3a^2b + 3ab^2 + b^3 = a^2(a+3b) + b^2(3a+b) \\\\ a^3 - 3a^2b + 3ab^2 - b^3 = a^2(a-3b) + b^2(3a-b) $$","blockMath-bfQIfRGGj",{"__TSPROSE_proseElement":219,"schema":8412,"children":8413},{"name":269,"type":222,"linkable":223,"__TSPROSE_schema":219},[8414,8439,8445],{"__TSPROSE_proseElement":219,"schema":8415,"children":8416,"id":8438},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8417,8420,8426,8429,8435],{"__TSPROSE_proseElement":219,"schema":8418,"data":8419},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Разбейте скобки вида ",{"__TSPROSE_proseElement":219,"schema":8421,"data":8422,"storageKey":8424,"id":8425},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8423},"a \\pm 3b","$ a \\pm 3b $","inlinerMath-qItzIQZbe",{"__TSPROSE_proseElement":219,"schema":8427,"data":8428},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," так, чтобы получить в явном виде ",{"__TSPROSE_proseElement":219,"schema":8430,"data":8431,"storageKey":8433,"id":8434},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8432},"a \\pm b","$ a \\pm b $","inlinerMath-Tz0eFZwYx",{"__TSPROSE_proseElement":219,"schema":8436,"data":8437},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," в каждой из них:","paragraph-u29RqoddG",{"__TSPROSE_proseElement":219,"schema":8440,"data":8441,"storageKey":8443,"id":8444},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8442,"freeze":223},"a^2(a+3b) + b^2(3a+b) = a^2((a+b) + 2b) + b^2((a+b) + 2a) \\\\ a^2(a-3b) + b^2(3a-b) = a^2((a-b) - 2b) + b^2((a-b) - 2a)","$$ a^2(a+3b) + b^2(3a+b) = a^2((a+b) + 2b) + b^2((a+b) + 2a) \\\\ a^2(a-3b) + b^2(3a-b) = a^2((a-b) - 2b) + b^2((a-b) - 2a) $$","blockMath-XpWmXZo89",{"__TSPROSE_proseElement":219,"schema":8446,"children":8447,"id":8459},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8448,8451,8457],{"__TSPROSE_proseElement":219,"schema":8449,"data":8450},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"После этого раскройте скобки и вынесите за скобки общий множитель ",{"__TSPROSE_proseElement":219,"schema":8452,"data":8453,"storageKey":8455,"id":8456},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8454},"(a \\pm b)","$ (a \\pm b) $","inlinerMath-QO1tCOnny",{"__TSPROSE_proseElement":219,"schema":8458,"data":295},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"paragraph-xlwQ5JYIn",{"__TSPROSE_proseElement":219,"schema":8461,"children":8462},{"name":3662,"type":222,"linkable":223,"__TSPROSE_schema":219},[8463,8482,8488,8501,8507,8514,8520,8541,8547,8560,8566,8573,8579],{"__TSPROSE_proseElement":219,"schema":8464,"children":8465,"id":8481},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8466,8469,8473,8475,8479],{"__TSPROSE_proseElement":219,"schema":8467,"data":8468},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Будем проводить вывод для формулы куба суммы. 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исходных чисел, без квадратов:","paragraph-D9YIfnCqd",{"__TSPROSE_proseElement":219,"schema":8821,"data":8822,"storageKey":8824,"id":8825},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8823,"freeze":223},"438271606 + 561728395 = 1000000001 \\\\ 561728395 - 438271606 = 123456789","$$ 438271606 + 561728395 = 1000000001 \\\\ 561728395 - 438271606 = 123456789 $$","blockMath-MCZhuA95A",{"__TSPROSE_proseElement":219,"schema":8827,"children":8828,"id":8832},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8829],{"__TSPROSE_proseElement":219,"schema":8830,"data":8831},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Умножить эти два числа друг на друга на самом деле очень просто:","paragraph-5bqA3fEQB",{"__TSPROSE_proseElement":219,"schema":8834,"data":8835,"storageKey":8837,"id":8838},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8836,"freeze":223},"(10000000000 + 1) \\cdot 123456789 = 123456789 \\cdot 1000000000 + 123456789 = 123456789000000000 + 123456789 = 123456789123456789","$$ (10000000000 + 1) \\cdot 123456789 = 123456789 \\cdot 1000000000 + 123456789 = 123456789000000000 + 123456789 = 123456789123456789 $$","blockMath-KeJbea51E",{"__TSPROSE_proseElement":219,"schema":8840,"children":8841,"id":8852},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8842,8845,8849],{"__TSPROSE_proseElement":219,"schema":8843,"data":8844},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"По ",{"__TSPROSE_proseElement":219,"schema":8846,"data":8847,"storageKey":3963,"id":8848},{"name":279,"type":249,"linkable":219,"__TSPROSE_schema":219},{"label":1346},"dependencyInliner-COnHycFH6-15",{"__TSPROSE_proseElement":219,"schema":8850,"data":8851},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219}," получается следующая ситуация:","paragraph-WmvT4yrvW",{"__TSPROSE_proseElement":219,"schema":8854,"data":8855,"storageKey":8857,"id":8858},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8856,"freeze":223},"123456789123456789 = 561728395^2 - 438271606^2","$$ 123456789123456789 = 561728395^2 - 438271606^2 $$","blockMath-qeXZO4FTi",{"__TSPROSE_proseElement":219,"schema":8860,"children":8861,"id":8871},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8862,8865,8869],{"__TSPROSE_proseElement":219,"schema":8863,"data":8864},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Теперь легко найти и само число ",{"__TSPROSE_proseElement":219,"schema":8866,"data":8867,"storageKey":8760,"id":8868},{"name":3672,"type":249,"linkable":219,"__TSPROSE_schema":219},{"katex":8759},"inlinerMath-9lqCBTu25-1",{"__TSPROSE_proseElement":219,"schema":8870,"data":3886},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"paragraph-F184pvKyh",{"__TSPROSE_proseElement":219,"schema":8873,"data":8874,"storageKey":8876,"id":8877},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8875,"freeze":223},"561728395^2 = 123456789123456789 + 438271606^2 = 123456789123456789 + 192082000625819236 = 315538789749275025","$$ 561728395^2 = 123456789123456789 + 438271606^2 = 123456789123456789 + 192082000625819236 = 315538789749275025 $$","blockMath-nzQSRTeqp",{"__TSPROSE_proseElement":219,"schema":8879,"children":8880,"id":8884},{"name":244,"type":222,"linkable":219,"__TSPROSE_schema":219},[8881],{"__TSPROSE_proseElement":219,"schema":8882,"data":8883},{"name":248,"type":249,"linkable":223,"__TSPROSE_schema":219},"Осталось найти сумму цифр получившегося числа:","paragraph-Jrq0XFoaj",{"__TSPROSE_proseElement":219,"schema":8886,"data":8887,"storageKey":8889,"id":8890},{"name":254,"type":222,"linkable":219,"__TSPROSE_schema":219},{"katex":8888,"freeze":223},"3 + 1 + 5 + 5 + 3 + 8 + 7 + 8 + 9 + 7 + 4 + 9 + 2 + 7 + 5 + 0 + 2 + 5 = \\boxed{91}","$$ 3 + 1 + 5 + 5 + 3 + 8 + 7 + 8 + 9 + 7 + 4 + 9 + 2 + 7 + 5 + 0 + 2 + 5 = \\boxed{91} $$","blockMath-B1CYTnagT","sverkhdlinnye-chisla",{"$$ x^2 + 2xy + y^2 $$":8893,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$squareSumFactorExamples":8894,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$squareDiffExamples":8901,"$$ (x + y)^2 $$":8893,"$$ (x - 2)^2 $$":8893,"$$ x^2 - 4x + 4 $$":8893,"$$ a^2 + 2a + 1 $$":8893,"$$ (a + 1)^2 $$":8893,"$$ (a + 11)^2 $$":8893,"$$ a^2 + 22a + 121 $$":8893,"$$ n^2 + m^2 + 2mn $$":8893,"$$ (n + m)^2 $$":8893,"$$ \\left(\\frac{1}{5} - m\\right)^2 $$":8893,"$$ \\frac{1}{25} - \\frac{2m}{5} + m^2 $$":8893,"$$ x^2 - 2xy + y^2 $$":8893,"$$ (x - y)^2 $$":8893,"$$ \\left(x + \\frac{2}{3}\\right)^2 $$":8893,"$$ x^2 + \\frac{4}{3}x + \\frac{4}{9} $$":8893,"$$ 81 - 18a + a^2 $$":8893,"$$ (9 - a)^2 $$":8893,"$$ (2x - 3)^2 $$":8893,"$$ 4x^2 - 12x + 9 $$":8893,"$$ a^2 - 12a + 36 $$":8893,"$$ (a - 6)^2 $$":8893,"$$ (3a + 5)^2 $$":8893,"$$ 9a^2 + 30a + 25 $$":8893,"$$ 4b^2 - 4b + 1 $$":8893,"$$ (2b - 1)^2 $$":8893,"$$ (-8 - 5b)^2 $$":8893,"$$ 64 + 80b + 25b^2 $$":8893,"$$ 1 + 10x + 25x^2 $$":8893,"$$ (1 + 5x)^2 $$":8893,"$$ (3x - 10y)^2 $$":8893,"$$ 9x^2 - 60xy + 100y^2 $$":8893,"$$ 100a^2 + 20a + 1 $$":8893,"$$ (10a + 1)^2 $$":8893,"$$ (11b - 2d)^2 $$":8893,"$$ 121b^2 - 44bd + 4d^2 $$":8893,"$$ 1 - 18y + 81y^2 $$":8893,"$$ (1 - 9y)^2 $$":8893,"$$ (6m + 5n)^2 $$":8893,"$$ 36m^2 + 60mn + 25n^2 $$":8893,"$$ 9 - 12x + 4x^2 $$":8893,"$$ (3 - 2x)^2 $$":8893,"$$ \\left(b + \\frac{5}{12}c\\right)^2 $$":8893,"$$ b^2 + \\frac{5}{6}bc + \\frac{25}{144}c^2 $$":8893,"$$ 9x^2 + 30x + 25 $$":8893,"$$ (3x + 5)^2 $$":8893,"$$ (4x - 9y)^2 $$":8893,"$$ 16x^2 - 72xy + 81y^2 $$":8893,"$$ 9a^2 - 30am + 25m^2 $$":8893,"$$ (3a - 5m)^2 $$":8893,"$$ (5y + 2z)^2 $$":8893,"$$ 25y^2 + 20yz + 4z^2 $$":8893,"$$ 16m^2 + 24mn + 9n^2 $$":8893,"$$ (4m + 3n)^2 $$":8893,"$$ \\left(-7y + \\frac{1}{7}x\\right)^2 $$":8893,"$$ 49y^2 - 2xy + \\frac{x^2}{49} $$":8893,"$$ 49 + 36x^2 + 84x $$":8893,"$$ (7 + 6x)^2 $$":8893,"$$ 4 + 49x^2 - 28x $$":8893,"$$ (2 - 7x)^2 $$":8893,"$$ 4x^2 + 36xy + 81y^2 $$":8893,"$$ (2x + 9y)^2 $$":8893,"$$ 1 - a^2 $$":8893,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$diffOfSquaresExamples":8907,"$$ (1 + a)(1 - a) $$":8893,"$$ (8 - a)(8 + a) $$":8893,"$$ 64 - a^2 $$":8893,"$$ 16 - x^2 $$":8893,"$$ (4 + x)(4 - x) $$":8893,"$$ (b - 5)(b + 5) $$":8893,"$$ b^2 - 25 $$":8893,"$$ -y^2 + 81 $$":8893,"$$ (9 + y)(9 - y) $$":8893,"$$ (7 + b)(7 - b) $$":8893,"$$ 49 - b^2 $$":8893,"$$ \\frac{1}{9} - b^2 $$":8893,"$$ \\left(\\frac{1}{3} + b\\right)\\left(\\frac{1}{3} - b\\right) $$":8893,"$$ (y + 2)(2 - y) $$":8893,"$$ 4 - y^2 $$":8893,"$$ y^2 - \\frac{25}{36} $$":8893,"$$ \\left(y + \\frac{5}{6}\\right)\\left(y - \\frac{5}{6}\\right) $$":8893,"$$ (a - 4)(4 + a) $$":8893,"$$ a^2 - 16 $$":8893,"$$ 0.81 - x^2 $$":8893,"$$ (0.9 + x)(0.9 - x) $$":8893,"$$ (1 + c)(-1 + c) $$":8893,"$$ c^2 - 1 $$":8893,"$$ 16a^2 - 1 $$":8893,"$$ (4a + 1)(4a - 1) $$":8893,"$$ (6 - a)(-a - 6) $$":8893,"$$ a^2 - 36 $$":8893,"$$ 100 - 9y^2 $$":8893,"$$ (10 + 3y)(10 - 3y) $$":8893,"$$ (-b + 1)(b + 1) $$":8893,"$$ 1 - b^2 $$":8893,"$$ 36a^2 - 25b^2 $$":8893,"$$ (6a + 5b)(6a - 5b) $$":8893,"$$ (-x - 5)(x - 5) $$":8893,"$$ 25 - x^2 $$":8893,"$$ -9p^2 + 0.16q^2 $$":8893,"$$ (0.4q + 3p)(0.4q - 3p) $$":8893,"$$ (1 - 4x)(1 + 4x) $$":8893,"$$ 1 - 16x^2 $$":8893,"$$ \\frac{64}{9}n^2 - 4m^2 $$":8893,"$$ \\left(\\frac{8}{3}n + 2m\\right)\\left(\\frac{8}{3}n - 2m\\right) $$":8893,"$$ (5b - 7)(5b + 7) $$":8893,"$$ 25b^2 - 49 $$":8893,"$$ 0.04x^2 - 0.64y^2 $$":8893,"$$ (0.2x + 0.8y)(0.2x - 0.8y) $$":8893,"$$ (8x + 9)(9 - 8x) $$":8893,"$$ 81 - 64x^2 $$":8893,"$$ 9m^2n^2 - 1 $$":8893,"$$ (3mn + 1)(3mn - 1) $$":8893,"$$ (4y + 5b)(4y - 5b) $$":8893,"$$ 16y^2 - 25b^2 $$":8893,"$$ 81 - 16p^2q^2 $$":8893,"$$ (9 + 4pq)(9 - 4pq) $$":8893,"$$ (0.5m + 7x)(7x - 0.5m) $$":8893,"$$ 49x^2 - 0.25m^2 $$":8893,"$$ 0.01a^2b^2 - 100m^2 $$":8893,"$$ (0.1ab + 10m)(0.1ab - 10m) $$":8893,"$$ \\left(\\frac{9}{7}y - 0.2z\\right)\\left(\\frac{9}{7}y + 0.2z\\right) $$":8893,"$$ \\frac{81}{49}y^2 - 0.04z^2 $$":8893,"$$ 121a^2b^4 - 49c^2 $$":8893,"$$ (11ab^2 + 7c)(11ab^2 - 7c) $$":8893,"$$ \\left(0.8ab - \\frac{1}{3}c\\right)\\left(0.8ab + \\frac{1}{3}c\\right) $$":8893,"$$ 0.64a^2b^2 - \\frac{c^2}{9} $$":8893,"$$ -\\frac{9}{64}t^2 + 36k^4l^6 $$":8893,"$$ \\left(6k^2l^3 + \\frac{3}{8}t\\right)\\left(6k^2l^3 - \\frac{3}{8}t\\right) $$":8893,"$$ (-0.4mn + 1.1xy)(0.4mn + 1.1xy) $$":8893,"$$ 1.21x^2y^2 - 0.16m^2n^2 $$":8893,"$$ \\left(-\\frac{1}{5}pq - \\frac{2}{9}q\\right)\\left(\\frac{1}{5}pq - \\frac{2}{9}q\\right) $$":8893,"$$ \\frac{4q^2}{81} - \\frac{p^2q^2}{25} $$":8893,"$$ (m + n)^3 $$":8893,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$cubeSumDiffExamples":8913,"$$ m^3 + 3m^2n + 3mn^2 + n^3 $$":8893,"$$ (c - d)^3 $$":8893,"$$ c^3 - 3c^2d + 3cd^2 - d^3 $$":8893,"$$ a^3 + 3a^2b + 3ab^2 + b^3 $$":8893,"$$ (a + b)^3 $$":8893,"$$ (2 + b)^3 $$":8893,"$$ 8 + 12b + 6b^2 + b^3 $$":8893,"$$ (3 - a)^3 $$":8893,"$$ 27 - 27a + 9a^2 - a^3 $$":8893,"$$ x^3 - 3x^2y + 3xy^2 - y^3 $$":8893,"$$ (x - y)^3 $$":8893,"$$ (x - 2)^3 $$":8893,"$$ x^3 - 6x^2 + 12x - 8 $$":8893,"$$ (a + 2b)^3 $$":8893,"$$ a^3 + 6a^2b + 12ab^2 + 8b^3 $$":8893,"$$ 125m^3 + 75m^2 + 15m + 1 $$":8893,"$$ (5m + 1)^3 $$":8893,"$$ (c - 3d)^3 $$":8893,"$$ c^3 - 9c^2d + 27cd^2 - 27d^3 $$":8893,"$$ \\left(4m + \\frac{1}{3}k\\right)^3 $$":8893,"$$ 64m^3 + 16m^2k + \\frac{4mk^2}{3} + \\frac{k^3}{27} $$":8893,"$$ 64 - 96a + 48a^2 - 8a^3 $$":8893,"$$ (4 - 2a)^3 $$":8893,"$$ \\left(\\frac{2}{3}a - 3b\\right)^3 $$":8893,"$$ \\frac{8a^3}{27} - 4a^2b + 18ab^2 - 27b^3 $$":8893,"$$ (a^2 + x^2)^3 $$":8893,"$$ a^6 + 3a^4x^2 + 3a^2x^4 + x^6 $$":8893,"$$ m^3 - 12m^2 + 48m - 64 $$":8893,"$$ (m - 4)^3 $$":8893,"$$ (2a^2 - 3b^2)^3 $$":8893,"$$ 8a^6 - 36a^4b^2 + 54a^2b^4 - 27b^6 $$":8893,"$$ (2x^3 - 3y^2)^3 $$":8893,"$$ 8x^9 - 36x^6y^2 + 54x^3y^4 - 27y^6 $$":8893,"$$ a^3 + 18a^2 + 108a + 216 $$":8893,"$$ (a + 6)^3 $$":8893,"$$ (10a^4 - 6b^2)^3 $$":8893,"$$ 1000a^{12} - 1800a^8b^2 + 1080a^4b^4 - 216b^6 $$":8893,"$$ \\left(\\frac{1}{2}a - \\frac{1}{3}b\\right)^3 $$":8893,"$$ \\frac{a^3}{8} - \\frac{a^2b}{4} + \\frac{ab^2}{6} - \\frac{b^3}{27} $$":8893,"$$ \\frac{a^3}{8} + \\frac{3a^2}{2} + 6a + 8 $$":8893,"$$ \\left(\\frac{a}{2} + 2\\right)^3 $$":8893,"$$ (0.5m + 0.1n)^3 $$":8893,"$$ 0.125m^3 + 0.075m^2n + 0.015mn^2 + 0.001n^3 $$":8893,"$$ \\frac{x^3}{27} - \\frac{x^2}{3} + x - 1 $$":8893,"$$ \\left(\\frac{x}{3} - 1\\right)^3 $$":8893,"$$ (2x - 3xy)^3 $$":8893,"$$ 8x^3 - 36x^3y + 54x^3y^2 - 27x^3y^3 $$":8893,"$$ \\frac{p^3}{8} + \\frac{9p^2q}{8} + \\frac{27pq^2}{8} + \\frac{27q^3}{8} $$":8893,"$$ \\left(\\frac{p}{2} + \\frac{3q}{2}\\right)^3 $$":8893,"$$ \\left(2a + \\frac{1}{2}ab\\right)^3 $$":8893,"$$ 8a^3 + 6a^3b + \\frac{3a^3b^2}{2} + \\frac{a^3b^3}{8} $$":8893,"$$ 8x^3 + 6x^2y + \\frac{3xy^2}{2} + \\frac{y^3}{8} $$":8893,"$$ \\left(2x + \\frac{y}{2}\\right)^3 $$":8893,"$$ \\left(0.1n^4 - \\frac{1}{2}n^3\\right)^3 $$":8893,"$$ 0.001n^{12} - 0.015n^{11} + 0.075n^{10} - 0.125n^9 $$":8893,"$$ \\frac{27a^3}{8} - \\frac{9a^2b}{2} + 2ab^2 - \\frac{8b^3}{27} $$":8893,"$$ \\left(\\frac{3a}{2} - \\frac{2b}{3}\\right)^3 $$":8893,"$$ (0.2x^2 - 0.3x^3)^3 $$":8893,"$$ 0.008x^6 - 0.036x^7 + 0.054x^8 - 0.027x^9 $$":8893,"$$ \\frac{a^3}{64} - \\frac{3a^2b}{32} + \\frac{3ab^2}{16} - \\frac{b^3}{8} $$":8893,"$$ \\left(\\frac{a}{4} - \\frac{b}{2}\\right)^3 $$":8893,"$$ 27 - 9a + a^2 - \\frac{a^3}{27} $$":8893,"$$ \\left(3 - \\frac{a}{3}\\right)^3 $$":8893,"$$ \\frac{x^3}{8} + \\frac{x^2y}{4} + \\frac{xy^2}{6} + \\frac{y^3}{27} $$":8893,"$$ \\left(\\frac{x}{2} + \\frac{y}{3}\\right)^3 $$":8893,"$$ (3x^2 + \\text{?})^2 = \\text{?} + \\text{?} + 16y^4 $$":8893,"$$ (3x^2 + 4y^2)^2 = 9x^4 + 24x^2y^2 + 16y^4 $$":8893,"$ 16y^4 $":8893,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$squareSum":8919,"$ b^2 $":8893,"$$ 16y^4 = (4y^2)^2 $$":8893,"$ 4y^2 $":8893,"$$ (3x^2 + 4y^2)^2 = (3x^2)^2 + 2 \\cdot (3x^2) \\cdot (4y^2) + (4y^2)^2 = 9x^4 + 24x^2y^2 + 16y^4 $$":8893,"$$ (\\text{?} - 5b^3)^2 = 49a^6 - \\text{?} + \\text{?} $$":8893,"$$ (7a^3 - 5b^3)^2 = 49a^6 - 70a^3b^3 + 25b^6 $$":8893,"$ a^2 $":8893,"$$ 49a^6 = (7a^3)^2 $$":8893,"$ 7a^3 $":8893,"$$ (7a^3 - 5b^3)^2 = (7a^3)^2 - 2 \\cdot (7a^3) \\cdot (5b^3) + (5b^3)^2 = 49a^6 - 70a^3b^3 + 25b^6 $$":8893,"$$ (4a^3 + \\text{?})^2 = \\text{?} + 40a^3b^4 + \\text{?} $$":8893,"$$ (4a^3 + 5b^4)^2 = 16a^6 + 40a^3b^4 + 25b^8 $$":8893,"$ 2ab $":8893,"$ 40a^3b^4 $":8893,"$ b $":8893,"$$ 40a^3b^4 = 2 \\cdot 20a^3b^4 = 2 \\cdot \\underset{a}{(4a^3)} \\cdot \\underset{b}{(5b^4)} $$":8893,"$ 5b^4 $":8893,"$$ (4a^3 + 5b^4)^2 = (4a^3)^2 + 2 \\cdot (4a^3) \\cdot (5b^4) + (5b^4)^2 = 16a^6 + 40a^3b^4 + 25b^8 $$":8893,"$$ (\\text{?} - 12a)(\\text{?} + \\text{?}) = 9b^2 - \\text{?} $$":8893,"$$ (3b - 12a)(3b + 12a) = 9b^2 - 144a^2 $$":8893,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$diffOfSquares":8922,"$ a^2 - b^2 $":8893,"$ 9b^2 $":8893,"$$ 9b^2 = (3b)^2 $$":8893,"$ 3b $":8893,"$ (3b + 12a) $":8893,"$ b^2 = (12a)^2 = 144a^2 $":8893,"$$ (3b - 12a)(3b + 12a) = (3b)^2 - (12a)^2 = 9b^2 - 144a^2 $$":8893,"$$ (\\text{?} - 5c)(\\text{?} + 5c) = 16d^2 - \\text{?} $$":8893,"$$ (4d - 5c)(4d + 5c) = 16d^2 - 25c^2 $$":8893,"$ (\\text{?} + 5c) $":8893,"$ 5c $":8893,"$ a^2 = 16d^2 $":8893,"$$ 16d^2 = (4d)^2 $$":8893,"$ 4d $":8893,"$ (5c)^2 = 25c^2 $":8893,"$$ (4d - 5c)(4d + 5c) = (4d)^2 - (5c)^2 = 16d^2 - 25c^2 $$":8893,"$$ (0{,}7p + \\text{?})(\\text{?} - 0{,}7p) = \\frac{1}{9}m^8 - 0{,}49p^2 $$":8893,"$$ \\left(0{,}7p + \\frac{1}{3}m^4\\right)\\left(\\frac{1}{3}m^4 - 0{,}7p\\right) = \\frac{1}{9}m^8 - 0{,}49p^2 $$":8893,"$ (b + a)(a - b) = a^2 - b^2 $":8893,"$ 0{,}7p $":8893,"$ a $":8893,"$$ \\frac{1}{9}m^8 = \\left(\\frac{1}{3}m^4\\right)^2 $$":8893,"$ (0{,}7p)^2 = 0{,}49p^2 $":8893,"$ a = \\frac{1}{3}m^4 $":8893,"$$ \\left(0{,}7p + \\frac{1}{3}m^4\\right)\\left(\\frac{1}{3}m^4 - 0{,}7p\\right) = \\left(\\frac{1}{3}m^4\\right)^2 - (0{,}7p)^2 = \\frac{1}{9}m^8 - 0{,}49p^2 $$":8893,"$$ (3m^2 + \\text{?})(\\text{?} - \\text{?}) = 9m^4 - n^6 $$":8893,"$$ (3m^2 + n^3)(3m^2 - n^3) = 9m^4 - n^6 $$":8893,"$ 3m^2 $":8893,"$$ 9m^4 = (3m^2)^2 $$":8893,"$ n^6 = (n^3)^2 $":8893,"$ b = n^3 $":8893,"$$ (3m^2 + n^3)(3m^2 - n^3) = (3m^2)^2 - (n^3)^2 = 9m^4 - n^6 $$":8893,"$$ (5a^4 + \\text{?})^3 = \\text{?} + \\text{?} + \\text{?} + 8b^{12} $$":8893,"$$ (5a^4 + 2b^4)^3 = 125a^{12} + 150a^8b^4 + 60a^4b^8 + 8b^{12} $$":8893,"$ 8b^{12} $":8893,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$cubeSum":8928,"$ b^3 $":8893,"$$ 8b^{12} = (2b^4)^3 $$":8893,"$ 2b^4 $":8893,"$$ (5a^4 + 2b^4)^3 = (5a^4)^3 + 3 \\cdot (5a^4)^2 \\cdot (2b^4) + 3 \\cdot (5a^4) \\cdot (2b^4)^2 + (2b^4)^3 = 125a^{12} + 150a^8b^4 + 60a^4b^8 + 8b^{12} $$":8893,"$$ (\\text{?} - 2x)^3 = z^6 - \\text{?} + \\text{?} - \\text{?} $$":8893,"$$ (z^2 - 2x)^3 = z^6 - 6xz^4 + 12x^2z^2 - 8x^3 $$":8893,"$ a^3 $":8893,"$$ z^6 = (z^2)^3 $$":8893,"$ z^2 $":8893,"$$ (z^2 - 2x)^3 = (z^2)^3 - 3 \\cdot (z^2)^2 \\cdot (2x) + 3 \\cdot (z^2) \\cdot (2x)^2 - (2x)^3 = z^6 - 6xz^4 + 12x^2z^2 - 8x^3 $$":8893,"$$ (\\text{?} + 4)^3 = \\text{?} + \\text{?} + 240a^5 + \\text{?} $$":8893,"$$ (5a^5 + 4)^3 = 125a^{15} + 300a^{10} + 240a^5 + 64 $$":8893,"$ 3ab^2 $":8893,"$ b = 4 $":8893,"$ 240a^5 $":8893,"$$ 240a^5 = 3 \\cdot 80a^5 = 3 \\cdot \\underset{a}{(5a^5)} \\cdot \\underset{b^2}{4^2} $$":8893,"$ 5a^5 $":8893,"$$ (5a^5 + 4)^3 = (5a^5)^3 + 3 \\cdot (5a^5)^2 \\cdot 4 + 3 \\cdot (5a^5) \\cdot 4^2 + 4^3 = 125a^{15} + 300a^{10} + 240a^5 + 64 $$":8893,"problemScript:content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-diff-of-squares":8934,"$ 91^2 - 90^2 $":8893,"$ 181 $":8893,"$$ 91^2 - 90^2 = (91 + 90)(91 - 90) = 181 \\cdot 1 = 181 $$":8893,"$ 1 $":8893,"$$ (a+b)^2 = a^2 + 2ab + b^2 \\\\ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$":8893,"$ (a-b)^2 $":8893,"$ (a-b)^3 $":8893,"$$ (a-b)^2 = (a + (-b))^2 >>{big} (a-b)^3 = (a + (-b))^3 $$":8893,"$$ (a + (-b))^2 = a^2 + 2a(-b) + (-b)^2 = \\boxed{a^2 - 2ab + b^2} \\\\ (a + (-b))^3 = a^3 + 3a^2(-b) + 3a(-b)^2 + (-b)^3 = \\boxed{a^3 - 3a^2b + 3ab^2 - b^3} $$":8893,"$ \\blacksquare $":8893,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$squareDiff":8936,"$$ (-a+b)^2 = \\text{?} $$":8893,"$ -a $":8893,"$$ ((-a) + b)^2 = (-a)^2 + 2 \\cdot (-a) \\cdot b + b^2 = \\boxed{a^2 - 2ab + b^2} $$":8893,"$ -2ab $":8893,"$$ (-a - b)^2 = \\text{?} $$":8893,"$ -b $":8893,"$$ ((-a) + (-b))^2 = (-a)^2 + 2 \\cdot (-a) \\cdot (-b) + (-b)^2 = \\boxed{a^2 + 2ab + b^2} $$":8893,"$$ (-a - b)^2 = (-(a + b))^2 = (-1)^2(a + b)^2 = a^2 + 2ab + b^2 $$":8893,"$ n $":8893,"$ n+1 $":8893,"$$ (n+1)^2 - n^2 = (\\cancel{n} + 1 - \\cancel{n})(n + 1 + n) = 1 \\cdot ((n+1) + n) = (n+1) + n $$":8893,"$ 4 $":8893,"$ 2n $":8893,"$ 2n + 2 $":8893,"$$ (2n + 2)^2 - (2n)^2 = (\\cancel{2n} + 2 - \\cancel{2n})(2n + 2 + 2n) = 2 \\cdot (4n + 2) = 4 \\cdot (2n + 1) $$":8893,"$ n^2 + (n+1)^2 $":8893,"$ 2n(n+1) $":8893,"$$ n^2 + (n+1)^2 - 2n(n+1) $$":8893,"$$ n^2 - 2n(n+1) + (n+1)^2 = (n - (n+1))^2 = (-1)^2 = 1 $$":8893,"problemScript:content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-product":8939,"$ 201 \\cdot 199 $":8893,"$ 39999 $":8893,"$ 201 = 200 + 1 $":8893,"$ 199 = 200 - 1 $":8893,"$$ 201 \\cdot 199 = (200 + 1)(200 - 1) = 200^2 - 1^2 = 40000 - 1 = 39999 $$":8893,"problemScript:content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-difference":8941,"$ 43440^2 - 43438 \\cdot 43442 $":8893,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fpractice\u002F$simpleProduct":8943,"$ 43438 = 43440 - 2 $":8893,"$ 43442 = 43440 + 2 $":8893,"$ 43440 $":8893,"$$ 43438 \\cdot 43442 = (43440 - 2)(43440 + 2) = 43440^2 - 2^2 $$":8893,"$$ 43440^2 - 43438 \\cdot 43442 = 43440^2 - \\left(43440^2 - 2^2\\right) = \\cancel{43440^2} - \\cancel{43440^2} + 2^2 = 2^2 = 4 $$":8893,"$ 43440^2 $":8893,"$ 2^2 = 4 $":8893,"$$ (\\sqrt5 + \\sqrt6 + \\sqrt7)(\\sqrt5 + \\sqrt6 - \\sqrt7)(\\sqrt5 - \\sqrt6 + \\sqrt7)(-\\sqrt5 + \\sqrt6 + \\sqrt7) $$":8893,"$ 104 $":8893,"$$ ((\\sqrt5 + \\sqrt6) + \\sqrt7)((\\sqrt5 + \\sqrt6) - \\sqrt7) = \\\\ (\\sqrt5 + \\sqrt6)^2 - (\\sqrt7)^2 = \\\\ (5 + 2 \\cdot \\sqrt{30} + 6) - 7 = \\\\ 4 + 2 \\cdot \\sqrt{30} $$":8893,"$$ (\\sqrt5 - \\sqrt6 + \\sqrt7)(-\\sqrt5 + \\sqrt6 + \\sqrt7) = \\\\ (\\sqrt7 + (\\sqrt5 - \\sqrt6))(\\sqrt7 - (\\sqrt5 - \\sqrt6)) = \\\\ (\\sqrt7)^2 - (\\sqrt5 - \\sqrt6)^2 = \\\\ 7 - (5 - 2 \\cdot \\sqrt{30} + 6) = \\\\ -4 + 2 \\cdot \\sqrt{30} $$":8893,"$$ (4 + 2 \\cdot \\sqrt{30})(-4 + 2 \\cdot \\sqrt{30}) = \\\\ (2 \\cdot \\sqrt{30} + 4)(2 \\cdot \\sqrt{30} - 4) = \\\\ (2 \\cdot \\sqrt{30})^2 - 4^2 = \\\\ 4 \\cdot 30 - 16 = \\\\ 104 $$":8893,"$$ \\frac{4x^2 + 12x + 9}{4x^2 - 9} $$":8893,"$$ \\frac{2x + 3}{2x - 3} $$":8893,"$$ 4x^2 + 12x + 9 = (2x)^2 + 2 \\cdot (2x) \\cdot 3 + 3^2 = (2x + 3)^2 $$":8893,"$$ 4x^2 - 9 = (2x)^2 - 3^2 = (2x + 3)(2x - 3) $$":8893,"$ (2x+3) $":8893,"$$ \\frac{(2x + 3)^2}{(2x + 3)(2x - 3)} = \\frac{\\cancel{(2x + 3)}(2x + 3)}{\\cancel{(2x + 3)}(2x - 3)} = \\frac{2x + 3}{2x - 3} $$":8893,"$$ (x^2 + y^2)^2 - (x^2 - y^2)^2 $$":8893,"$$ 4x^2y^2 $$":8893,"$ x^2 + y^2 $":8893,"$ x^2 - y^2 $":8893,"$$ (x^2 + y^2)^2 - (x^2 - y^2)^2 = \\bigl((x^2+y^2) + (x^2-y^2)\\bigr)\\bigl((x^2+y^2) - (x^2-y^2)\\bigr) $$":8893,"$$ (x^2 + \\cancel{y^2} + x^2 - \\cancel{y^2})(\\cancel{x^2} + y^2 - \\cancel{x^2} + y^2) = 2x^2 \\cdot 2y^2 = 4x^2y^2 $$":8893,"$$ \\frac{a^2 - b^2}{a^2 + 2ab + b^2} $$":8893,"$$ \\frac{a - b}{a + b} $$":8893,"$$ \\frac{a^2 - b^2}{a^2 + 2ab + b^2} = \\frac{(a - b)(a + b)}{(a + b)^2} $$":8893,"$ (a+b) $":8893,"$$ \\frac{(a - b)\\cancel{(a + b)}}{\\cancel{(a + b)}(a + b)} = \\frac{a - b}{a + b} $$":8893,"$$ \\frac{x - \\dfrac{9}{x}}{1 - \\dfrac{3}{x}} $$":8893,"$$ x + 3 $$":8893,"$ x $":8893,"$$ \\frac{\\left(x - \\dfrac{9}{x}\\right) \\cdot x}{\\left(1 - \\dfrac{3}{x}\\right) \\cdot x} = \\frac{x^2 - 9}{x - 3} $$":8893,"$$ \\frac{x^2 - 9}{x - 3} = \\frac{(x + 3)\\cancel{(x - 3)}}{\\cancel{x - 3}} = x + 3 $$":8893,"$$ \\frac{a^2 + 2ab + b^2 - 9}{a + b + 3} $$":8893,"$$ a + b - 3 $$":8893,"$$ a^2 + 2ab + b^2 - 9 = (a + b)^2 - 9 $$":8893,"$$ (a + b)^2 - 9 = (a + b)^2 - 3^2 = (a + b + 3)(a + b - 3) $$":8893,"$ (a + b + 3) $":8893,"$$ \\frac{(a + b + 3)(a + b - 3)}{a + b + 3} = a + b - 3 $$":8893,"$$ (a^2 - b^2)^2 + 4a^2b^2 $$":8893,"$$ (a^2 + b^2)^2 $$":8893,"$ (a^2 - b^2)^2 $":8893,"$$ (a^2 - b^2)^2 = a^4 - 2a^2b^2 + b^4 $$":8893,"$$ a^4 - 2a^2b^2 + b^4 + 4a^2b^2 = a^4 + 2a^2b^2 + b^4 $$":8893,"$$ a^4 + 2a^2b^2 + b^4 = (a^2)^2 + 2 \\cdot a^2 \\cdot b^2 + (b^2)^2 = (a^2 + b^2)^2 $$":8893,"$$ \\frac{\\dfrac{a+b}{a-b} + \\dfrac{a-b}{a+b}}{\\dfrac{(a+b)^2 + (a-b)^2}{a^2 - b^2}} $$":8893,"$$ 1 $$":8893,"$ (a-b)(a+b) $":8893,"$$ \\frac{a+b}{a-b} + \\frac{a-b}{a+b} = \\frac{(a+b)^2 + (a-b)^2}{(a-b)(a+b)} $$":8893,"$$ (a+b)^2 + (a-b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2 $$":8893,"$$ \\frac{a+b}{a-b} + \\frac{a-b}{a+b} = \\frac{2a^2 + 2b^2}{a^2 - b^2} $$":8893,"$$ \\frac{\\dfrac{2a^2 + 2b^2}{a^2 - b^2}}{\\dfrac{(a+b)^2 + (a-b)^2}{a^2 - b^2}} = \\frac{2a^2 + 2b^2}{a^2 - b^2} \\cdot \\frac{a^2 - b^2}{(a+b)^2 + (a-b)^2} $$":8893,"$$ \\frac{\\cancel{2a^2 + 2b^2}}{\\cancel{a^2 - b^2}} \\cdot \\frac{\\cancel{a^2 - b^2}}{\\cancel{2a^2 + 2b^2}} = 1 $$":8893,"$$ 10^x = \\left( 10^{624} + 25 \\right)^2 - \\left( 10^{624} - 25 \\right)^2 $$":8893,"$ 10 $":8893,"$ 626 $":8893,"$$ \\left( 10^{624} + 25 \\right)^2 - \\left( 10^{624} - 25 \\right)^2 = \\\\ \\left( 10^{624} + \\cancel{25} + 10^{624} - \\cancel{25} \\right)\\left( \\cancel{10^{624}} + 25 - \\cancel{10^{624}} + 25 \\right) = \\\\ 2 \\cdot 10^{624} \\cdot 50 = 10^{624} \\cdot 100 = 10^{624} \\cdot 10^2 = 10^{626} $$":8893,"$$ 10^x = 10^{626} $$":8893,"$ x = 626 $":8893,"$$ \\left( 1 - \\frac{1}{2^2} \\right) \\left( 1 - \\frac{1}{3^2} \\right) \\left( 1 - \\frac{1}{4^2} \\right) \\cdots \\left( 1 - \\frac{1}{n^2} \\right) $$":8893,"$ n = 100 $":8893,"$$ \\frac{n+1}{2n} $$":8893,"$ n=100 $":8893,"$$ \\frac{101}{200} $$":8893,"$$ \\left( \\frac{2^2 - 1}{2^2} \\right) \\left( \\frac{3^2 - 1}{3^2} \\right) \\left( \\frac{4^2 - 1}{4^2} \\right) \\cdots \\left( \\frac{n^2 - 1}{n^2} \\right) $$":8893,"$$ \\left( \\frac{(2-1)(2+1)}{2^2} \\right) \\left( \\frac{(3-1)(3+1)}{3^2} \\right) \\left( \\frac{(4-1)(4+1)}{4^2} \\right) \\cdots \\left( \\frac{(n-1)(n+1)}{n^2} \\right) $$":8893,"$$ \\frac{1 \\cdot 3 \\cdot 2 \\cdot 4 \\cdot 3 \\cdot 5 \\cdots (n-1)(n+1)}{2^2 \\cdot 3^2 \\cdot 4^2 \\cdots n^2} $$":8893,"$$ \\frac{(\\cancel{1 \\cdot 2 \\cdot 3 \\cdots (n-1)})(\\cancel{3 \\cdot 4 \\cdot 5 \\cdots n} \\cdot (n+1))}{(2 \\cdot \\cancel{3 \\cdots (n-1)} \\cdot n) \\cdot (\\cancel{2 \\cdot 3 \\cdots n})} = \\frac{n+1}{2n} $$":8893,"$$ \\frac{n+1}{2n} = \\frac{100+1}{2 \\cdot 100} = \\frac{101}{200} $$":8893,"$$ 2^x - 2^y = 1 >>{big} 4^x - 4^y = \\frac{5}{3} >>{big} x - y = \\text{?} $$":8893,"$$ \\frac{a^{n}}{a^{m}} = a^{n-m} $$":8893,"$ 2 $":8893,"$$ \\frac{5}{3} = 4^x - 4^y = (2^2)^x - (2^2)^y = 2^{2x} - 2^{2y} = (2^x)^2 - (2^y)^2 $$":8893,"$$ \\frac{5}{3} = (2^x)^2 - (2^y)^2 = (2^x + 2^y)\\underbrace{(2^x - 2^y)}_{\\small 1} \\\\ 2^x + 2^y = \\frac{5}{3} \\\\ $$":8893,"$$ (2^x + \\cancel{2^y}) + (2^x - \\cancel{2^y}) = \\frac{5}{3} + 1 \\\\ 2 \\cdot 2^x = \\frac{8}{3} \\\\ 2^x = \\frac{4}{3} $$":8893,"$$ (2^x + \\cancel{2^y}) - (2^x - \\cancel{2^y}) = \\frac{5}{3} - 1 \\\\ 2 \\cdot 2^y = \\frac{2}{3} \\\\ 2^y = \\frac{1}{3} $$":8893,"$$ \\frac{2^x}{2^y} = 2^{x-y} = \\frac{\\frac{4}{3}}{\\frac{1}{3}} = 4 = 2^2 \\\\ 2^{x-y} = 2^2 \\\\ $$":8893,"$ x-y $":8893,"$$ x - y = 2 $$":8893,"$ 693 $":8893,"$ m $":8893,"$ 10 \\cdot n + m $":8893,"$$ 59 = 10 \\cdot 5 + 9 >>{big} 13 = 10 \\cdot 1 + 3 >>{big} 92 = 10 \\cdot 9 + 2 >>{big} $$":8893,"$ 43 $":8893,"$ 34 $":8893,"$ 10n + m $":8893,"$ 10m + n $":8893,"$$ (10n + m)^2 - (10m + n)^2 = 693 \\\\ (10n + m + 10m + n)(10n + m - 10m - n) = 693 \\\\ (11n + 11m)(9n - 9m) = 693 \\\\ 99(n + m)(n - m) = 693 $$":8893,"$ 99 $":8893,"$$ (n + m)(n - m) = \\frac{693}{99} = 7 \\\\ (n + m)(n - m) = 7 $$":8893,"$ n+m $":8893,"$ n-m $":8893,"$ 7 $":8893,"$$ n + m = 7 \\\\ n - m = 1 $$":8893,"$ 9 $":8893,"$ n=4 $":8893,"$ m=3 $":8893,"$$ a^3 \\pm b^3 = (a \\pm b)(a^2 \\mp ab \\pm b^2) $$":8893,"$$ (a + 1)(a^2 - a + 1) $$":8893,"$$ a^3 + 1 $$":8893,"$$ a^3 + 8 $$":8893,"$$ (a + 2)(a^2 - 2a + 4) $$":8893,"$$ (x - 2)(x^2 + 2x + 4) $$":8893,"$$ x^3 - 8 $$":8893,"$$ x^3 - 27 $$":8893,"$$ (x - 3)(x^2 + 3x + 9) $$":8893,"$$ (2a + 3)(4a^2 - 6a + 9) $$":8893,"$$ 8a^3 + 27 $$":8893,"$$ 1 + y^3 $$":8893,"$$ (1 + y)(1 - y + y^2) $$":8893,"$$ (3 - b)(9 + 3b + b^2) $$":8893,"$$ 27 - b^3 $$":8893,"$$ m^3 - 27 $$":8893,"$$ (m - 3)(m^2 + 3m + 9) $$":8893,"$$ (1 + m^2)(1 - m^2 + m^4) $$":8893,"$$ 1 + m^6 $$":8893,"$$ p^3 - 1 $$":8893,"$$ (p - 1)(p^2 + p + 1) $$":8893,"$$ (3a - 4)(9a^2 + 12a + 16) $$":8893,"$$ 27a^3 - 64 $$":8893,"$$ 1 - 8x^3 $$":8893,"$$ (1 - 2x)(1 + 2x + 4x^2) $$":8893,"$$ (3x - 4)(9x^2 + 12x + 16) $$":8893,"$$ 27x^3 - 64 $$":8893,"$$ 27 - 8a^3 $$":8893,"$$ (3 - 2a)(9 + 6a + 4a^2) $$":8893,"$$ \\left(a + \\frac{1}{2}\\right)\\left(a^2 - \\frac{1}{2}a + \\frac{1}{4}\\right) $$":8893,"$$ a^3 + \\frac{1}{8} $$":8893,"$$ 8y^3 + x^3 $$":8893,"$$ (2y + x)(4y^2 - 2xy + x^2) $$":8893,"$$ \\left(\\frac{1}{2}x - \\frac{1}{3}y\\right)\\left(\\frac{1}{4}x^2 + \\frac{1}{6}xy + \\frac{1}{9}y^2\\right) $$":8893,"$$ \\frac{x^3}{8} - \\frac{y^3}{27} $$":8893,"$$ x^6 - y^9 $$":8893,"$$ (x^2 - y^3)(x^4 + x^2 y^3 + y^6) $$":8893,"$$ \\left(\\frac{1}{2}a - 2b\\right)\\left(\\frac{1}{4}a^2 + ab + 4b^2\\right) $$":8893,"$$ \\frac{a^3}{8} - 8b^3 $$":8893,"$$ 27a^3 - 8y^3 $$":8893,"$$ (3a - 2y)(9a^2 + 6ay + 4y^2) $$":8893,"$$ (16 + 4x + x^2)(x - 4) $$":8893,"$$ x^3 - 64 $$":8893,"$$ 1000x^{12}y^3 + 0{,}001a^9b^{15} $$":8893,"$$ (10x^4y + 0{,}1a^3b^5)(100x^8y^2 - x^4ya^3b^5 + 0{,}01a^6b^{10}) $$":8893,"$$ (49 + b^2 - 7b)(b + 7) $$":8893,"$$ b^3 + 343 $$":8893,"$ 111 $":8893,"$ 507 $":8893,"$ 2k + 1 $":8893,"$ k $":8893,"$$ 7 = 4^2 - 3^2 >>{big} 111 = 56^2 - 55^2 >>{big} 507 = 254^2 - 253^2 $$":8893,"$ 2k $":8893,"$$ 2k + 1 = \\left[ (k+1) + k \\right] = \\ldots $$":8893,"$$ \\ldots = \\left[ (k+1) + k \\right] \\cdot \\brand{1} = \\ldots $$":8893,"$$ \\ldots = \\left[ (k+1) + k \\right] \\cdot \\underbrace{\\left[ (k+1) - k \\right]}_{\\small 1} $$":8893,"$$ 2k + 1 = \\left[ (k+1) + k \\right] \\cdot \\left[ (k+1) - k \\right] = (k+1)^2 - k^2 \\\\ 2k + 1 = (k+1)^2 - k^2 $$":8893,"$$ 7 = 2 \\cdot 3 + 1 \\Rightarrow 3 >>{big} 111 = 2 \\cdot 55 + 1 \\Rightarrow 55 >>{big} 507 = 2 \\cdot 253 + 1 \\Rightarrow 253 $$":8893,"$$ 7 = \\boxed{4^2 - 3^2} >>{big} 111 = \\boxed{56^2 - 55^2} >>{big} 507 = \\boxed{254^2 - 253^2} $$":8893,"$$ a^3 + 3a^2b + 3ab^2 + b^3 \\overset{\\text{?}}{\\implies} (a+b)^3 \\\\ a^3 - 3a^2b + 3ab^2 - b^3 \\overset{\\text{?}}{\\implies} (a-b)^3 $$":8893,"$$ a^3 + 3a^2b + 3ab^2 + b^3 = a^2(a+3b) + b^2(3a+b) \\\\ a^3 - 3a^2b + 3ab^2 - b^3 = a^2(a-3b) + b^2(3a-b) $$":8893,"$ a \\pm 3b $":8893,"$ a \\pm b $":8893,"$$ a^2(a+3b) + b^2(3a+b) = a^2((a+b) + 2b) + b^2((a+b) + 2a) \\\\ a^2(a-3b) + b^2(3a-b) = a^2((a-b) - 2b) + b^2((a-b) - 2a) $$":8893,"$ (a \\pm b) $":8893,"$$ a^2(a+3b) + b^2(3a+b) $$":8893,"$$ a^2((a+b) + 2b) + b^2((a+b) + 2a) $$":8893,"$$ a^2(a+b) + 2a^2b + b^2(a+b) + 2ab^2 $$":8893,"$$ (a+b)(a^2 + b^2) + 2ab(a+b) $$":8893,"$$ (a+b)(a^2 + 2ab + b^2) $$":8893,"$$ (a+b)(a+b)^2 = (a+b)^3 $$":8893,"$$ x = \\frac{4}{(\\sqrt{5} + 1)(\\sqrt[4]{5} + 1)(\\sqrt[8]{5} + 1)(\\sqrt[16]{5} + 1)} $$":8893,"$ (1+x)^{48} $":8893,"$ 125 $":8893,"$$ \\frac{4 \\yellow{(\\sqrt{5} - 1) (\\sqrt[4]{5} - 1) (\\sqrt[8]{5} - 1) (\\sqrt[16]{5} - 1) }}{(\\sqrt{5} + 1)\\yellow{(\\sqrt{5} - 1)}(\\sqrt[4]{5} + 1)\\yellow{(\\sqrt[4]{5} - 1)}(\\sqrt[8]{5} + 1)\\yellow{(\\sqrt[8]{5} - 1)}(\\sqrt[16]{5} + 1)\\yellow{(\\sqrt[16]{5} - 1)}} $$":8893,"$$ \\frac{\\cancel{4} \\cancel{(\\sqrt{5} - 1)} \\cancel{(\\sqrt[4]{5} - 1)} \\cancel{(\\sqrt[8]{5} - 1)} (\\sqrt[16]{5} - 1)}{\\cancel{(5 - 1)} \\cancel{(\\sqrt{5} - 1)}\\cancel{(\\sqrt[4]{5} - 1)}\\cancel{(\\sqrt[8]{5} - 1)}} = \\\\ \\sqrt[16]{5} - 1 $$":8893,"$$ (1+x)^{48} = (\\cancel{1} + \\sqrt[16]{5} - \\cancel{1})^{48} = (\\sqrt[16]{5})^{48} = 5^{\\frac{48}{16}} = 5^3 = 125 $$":8893,"$$ 438271606^2 = 192082000625819236 $$":8893,"$ 561728395^2 $":8893,"$ 91 $":8893,"$$ 438271606 + 561728395 = 1000000001 \\\\ 561728395 - 438271606 = 123456789 $$":8893,"$$ (10000000000 + 1) \\cdot 123456789 = 123456789 \\cdot 1000000000 + 123456789 = 123456789000000000 + 123456789 = 123456789123456789 $$":8893,"$$ 123456789123456789 = 561728395^2 - 438271606^2 $$":8893,"$$ 561728395^2 = 123456789123456789 + 438271606^2 = 123456789123456789 + 192082000625819236 = 315538789749275025 $$":8893,"$$ 3 + 1 + 5 + 5 + 3 + 8 + 7 + 8 + 9 + 7 + 4 + 9 + 2 + 7 + 5 + 0 + 2 + 5 = \\boxed{91} $$":8893},null,{"type":8895,"schemaName":227,"elementTitle":8896,"content":8897,"resolvedHref":8898,"previewRequest":8899},"unique","Запаковка в квадрат суммы",{"contentType":180,"contentTitle":159,"topicPart":216},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=square-sum-factor-examples",{"type":8895,"contentFullId":157,"contentType":180,"topicPart":216,"uniqueName":8900},"squareSumFactorExamples",{"type":8895,"schemaName":227,"elementTitle":8902,"content":8903,"resolvedHref":8904,"previewRequest":8905},"Примеры использования квадрата 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квадратов",{"contentType":180,"contentTitle":159,"topicPart":216},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=diff-of-squares",{"type":8895,"contentFullId":157,"contentType":180,"topicPart":216,"uniqueName":8927},"diffOfSquares",{"type":8895,"schemaName":188,"elementTitle":8929,"content":8930,"resolvedHref":8931,"previewRequest":8932},"Формула куба суммы",{"contentType":180,"contentTitle":159,"topicPart":216},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum",{"type":8895,"contentFullId":157,"contentType":180,"topicPart":216,"uniqueName":8933},"cubeSum",{"resolvedScriptSrc":8935},"\u002Fapi\u002FproblemScript\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-diff-of-squares.js",{"type":8895,"schemaName":188,"elementTitle":191,"content":8937,"resolvedHref":192,"previewRequest":8938},{"contentType":180,"contentTitle":159,"topicPart":216},{"type":8895,"contentFullId":157,"contentType":180,"topicPart":216,"uniqueName":190},{"resolvedScriptSrc":8940},"\u002Fapi\u002FproblemScript\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-product.js",{"resolvedScriptSrc":8942},"\u002Fapi\u002FproblemScript\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fsimple-difference.js",{"type":8895,"schemaName":4655,"elementTitle":5362,"content":8944,"resolvedHref":8945,"previewRequest":8946},{"contentType":180,"contentTitle":159,"topicPart":214},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=simple-product",{"type":8895,"contentFullId":157,"contentType":180,"topicPart":214,"uniqueName":8947},"simpleProduct",[8949,8956,8963,8968,8973,8979,8986,8990,8995,9001,9009,9017,9022,9027],{"link":8950,"schemaName":8951,"title":8952,"seo":8953,"description":8955},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=chto-takoe-fsu","heading","Что такое ФСУ?",{"title":8954},"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":8957,"schemaName":8958,"title":8959,"key":8960,"seo":8961,"description":8962},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=what-are-special-products","accent_term","Формулы сокращённого умножения",{"title":8952},{},"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":8964,"key":8965,"seo":8966,"description":8967},"Квадрат суммы",{},{},"Одна из формул сокращенного умножения (ФСУ): (a+b)² = a² + 2ab + b².\n          Позволяет быстро раскрывать скобки или же наоборот, запаковывать в них уже разложенные выражения.",{"link":192,"schemaName":188,"title":8969,"key":8970,"seo":8971,"description":8972},"Квадрат разности",{},{},"Одна из формул сокращенного умножения (ФСУ): (a-b)² = a² - 2ab + b².\n          Позволяет быстро раскрывать скобки или же наоборот, запаковывать в них уже разложенные выражения.",{"link":8974,"schemaName":8951,"title":8975,"seo":8976,"description":8978},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=primenenie-kvadrata-summy-ili-raznosti","Применение квадрата суммы или разности",{"title":8977},"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":8980,"schemaName":227,"title":8981,"key":8982,"seo":8983,"description":8985},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=fast-square","Быстрые квадраты",{},{"title":8984},"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":8925,"schemaName":188,"title":4697,"key":8987,"seo":8988,"description":8989},{},{},"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":8931,"schemaName":188,"title":8991,"key":8992,"seo":8993,"description":8994},"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":8996,"schemaName":188,"title":8997,"key":8998,"seo":8999,"description":9000},"\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":9002,"schemaName":9003,"title":9004,"key":9005,"seo":9006,"description":9008},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=memorization-tips","list","Как запомнить ФСУ?",{},{"title":9007},"Как запомнить формулы сокращенного умножения?","Набор советов и приемов, которые помогут наизусть запомнить формулы сокращенного умножения (ФСУ) и не путаться в них.",{"link":9010,"schemaName":9011,"title":9012,"key":9013,"seo":9015,"description":9016},"\u002Fsummary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=special-products-table","table","Таблица формул сокращенного умножения",{"title":9014},"Таблица ФСУ",{},"Единая таблица со всеми формулами сокращенного умножения.",{"link":9018,"schemaName":227,"title":9019,"seo":9020,"description":9021},"\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":9023,"schemaName":227,"title":9024,"seo":9025,"description":9026},"\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":9028,"schemaName":227,"title":9029,"seo":9030,"description":9031},"\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 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уравнения",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Felementary\u002F",[216,217,214],{"type":8,"shortId":9083,"title":9084,"flags":9085,"link":9086},"foundations\u002Fequations\u002Fzero-product-property","Нулевые множители",{},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fzero-product-property\u002F",{"type":9062,"separator":223,"shortId":9088,"title":9089,"flags":9090,"link":9091,"children":9092},"foundations\u002Fequations\u002Fquadratic","Квадратные уравнения",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002Fquadratic\u002F",[9093,9099,9105,9110,9116,9122,9127,9133,9139],{"type":180,"shortId":9094,"title":9095,"flags":9096,"link":9097,"parts":9098},"foundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it","Что это такое?",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it\u002F",[216,217,214],{"type":180,"shortId":9100,"title":9101,"flags":9102,"link":9103,"parts":9104},"foundations\u002Fequations\u002Fquadratic\u002Fincomplete","Неполная форма",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fincomplete\u002F",[216,217,214],{"type":180,"shortId":9106,"title":9107,"flags":9108,"link":182,"parts":9109},"foundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square","Полный квадрат",{},[216,217,214],{"type":180,"shortId":9111,"title":9112,"flags":9113,"link":9114,"parts":9115},"foundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula","Формула корней",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002F",[216,217,214],{"type":180,"shortId":9117,"title":9118,"flags":9119,"link":9120,"parts":9121},"foundations\u002Fequations\u002Fquadratic\u002Ffactoring","Разложение на множители",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002F",[216,217,214],{"type":8,"shortId":9123,"title":9124,"flags":9125,"link":9126},"foundations\u002Fequations\u002Fquadratic\u002Freal-world","В реальной жизни",{"secondary":219},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F",{"type":180,"shortId":9128,"title":9129,"flags":9130,"link":9131,"parts":9132},"foundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas","Теорема Виета",{"secondary":219},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas\u002F",[216,217,214],{"type":180,"shortId":9134,"title":9135,"flags":9136,"link":9137,"parts":9138},"foundations\u002Fequations\u002Fquadratic\u002Fmental-solving","Решение в уме",{"secondary":219,"advanced":219},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving\u002F",[216,217],{"type":180,"shortId":9140,"title":9141,"flags":9142,"link":9143,"parts":9144},"foundations\u002Fequations\u002Fquadratic\u002Fformulas","Общие формулы",{"secondary":219},"\u002Fsummary\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fformulas\u002F",[217,214],{"cameo":9146,"sponsor":9157},[9147,9148,9149,9150,9151,9152,9153,9154,9155,9156],"cameo-black-hole","cameo-confucius","cameo-dunno","cameo-einstein","cameo-euclid","cameo-gagarin","cameo-gauss","cameo-gorinich","cameo-newton","cameo-t800",[9158,9159],"sponsor-green-hedgehog","sponsor-reverse-shift",1777651272394]