Помогите решить уравнение! : (x - 1)(x - 7)(x - 4)( x + 2) = 40; Заранее спасибо!. Created by DenQ04a. algebra-ru

Помогите решить уравнение! : (x - 1)(x - 7)(x - 4)( x + 2) = 40; Заранее спасибо...

DenQ04a @DenQ04a

July 2022 1 43 Report

Помогите решить уравнение! :
(x - 1)(x - 7)(x - 4)( x + 2) = 40;
Заранее спасибо!

Answers & Comments

Dимасuk

Verified answer

(x - 1)(x - 4)(x - 7)(x + 2) = 40
(x² - 4x - x + 4)(x² + 2x - 7x - 14) = 40
(x² - 5x + 4)(x² - 5x - 14) = 40
Пусть t = x² - 5x + 4.
t(t - 14 - 4) = 40
t² - 18t = 40
t² - 18t + 81 - 81 - 40 = 0
(t - 9)² - 121 = 0
(t - 9)² - 11² = 0
(t - 9 - 11)(t - 9 + 11) = 0
(t - 20)(t + 2) = 0
t = -2; 20
Обратная замена:
1) x² - 5x + 4 = -2
x² - 5x + 6 = 0
x₁ + x₂ = 5
x₁x₂ = 6
x₁ = 2; x₂ = 3
2) x² - 5x + 4 = 20
x² - 5x -16 = 0
D = 25 + 16·4 = 89 = (√89)²
x₁ = (5 + √89)/2
x₂ = (5 - √89)/2
Ответ: x = (5 - √89)/2; 2; 3; (5 + √89)/2.

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