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Limbo206
@Limbo206
August 2022
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Пусть x1, x2 — корни уравнения 2x^2+3x−4=0. Чему равно значение выражения 16(x1^4+x2^4)?
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Удачник66
Verified answer
По теореме Виета
x1 + x2 = -b/a = -3/2
x1*x2 = c/a = -4/2 = -2
x1^4 + x2^4 = x1^4 + 2x1^2*x2^2 + x2^4 - 2x1^2*x2^2 =
= (x1^2 + x2^2)^2 - 2(x1*x2)^2 =
= (x1^2 + 2x1*x2 + x2^2 - 2x1*x2)^2 - 2(x1*x2)^2 =
= ((x1 + x2)^2 - 2x1*x2)^2 - 2(x1*x2)^2 = ((-3/2)^2 - 2(-2))^2 - 2(-2)^2 =
= (9/4 + 4)^2 - 2*4 = (25/4)^2 - 8 = 625/16 - 128/16 = 497/16
16(x1^4+x2^4) = 16*497/16 = 497
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Answers & Comments
Verified answer
По теореме Виетаx1 + x2 = -b/a = -3/2
x1*x2 = c/a = -4/2 = -2
x1^4 + x2^4 = x1^4 + 2x1^2*x2^2 + x2^4 - 2x1^2*x2^2 =
= (x1^2 + x2^2)^2 - 2(x1*x2)^2 =
= (x1^2 + 2x1*x2 + x2^2 - 2x1*x2)^2 - 2(x1*x2)^2 =
= ((x1 + x2)^2 - 2x1*x2)^2 - 2(x1*x2)^2 = ((-3/2)^2 - 2(-2))^2 - 2(-2)^2 =
= (9/4 + 4)^2 - 2*4 = (25/4)^2 - 8 = 625/16 - 128/16 = 497/16
16(x1^4+x2^4) = 16*497/16 = 497