Question

Решите уравнение x(x+1)(x+2)(x+3)=24 используя метод замены переменой

Answer 1

x(x + 1)(x + 2)(x + 3) = 24

[x(x + 3)] [(x + 1)(x + 2)] = 24

(x² + 3x)(x² + 2x + x + 2) = 24

(x² + 3x)(x² + 3x + 2) = 24

x² + 3x = t

t * (t + 2) = 24

t² + 2t - 24 = 0

D = 2² - 4 * ( - 24) = 4 + 96 = 100 = 10²

$t_{1}=\frac{-2+10}{2}=4\\\\t_{2}=\frac{-2-10}{2}=-6$

$1)x^{2}+3x= 4\\\\x^{2} +3x-4=0\\\\D=3^{2}-4*(-4)=9+16=25=5^{2}\\\\x_{1}=\frac{-3+5}{2}=1\\\\x_{2}=\frac{-3-5}{2}=-4\\\\2)x^{2} +3x=-6\\\\x^{2}+3x+6=0\\\\D=3^{2}-4*6=9-24=-15<0$

D < 0 - решений нет

Ответ : - 4 ; 1