Question

Решите уравнение 1) x³-x=0. 2) x⁴+x²=0. 3) x⁴-8x³. 4) 49x³+14x²+x=0. 5) x³+x²-x-1=0. 6) x³-4x²-25x+100=0

Answer 1

$x^3-x=0\\ x(x^2-1)=0\\\\ x=0\\\\ x^2-1=0\\ x^2=1\\ x=\pm1$
ответ: $x_1=0; \ x_2=-1; \ x_3=1$

$x^4+x^2=0\\ x^2(x^2+1)=0\\ x^2=0\\ x=0\\\\ x^2+1=0\\ x^2=-1\\ x=\varnothing$
ответ: $x=0$

$x^4-8x^3=0\\ x^3(x-8)=0\\\\ x^3=0\\ x=0\\\\ x-8=0\\ x=8$
ответ: $x_1=0; \ x_2=8$

$49x^3+14x^2+x=0\\ x(49x^2+14x+1)=0\\\\ x=0\\\\ 49x^2+14x+1=0\\ D=196-196=0\\\\ x= -\frac{14}{98}=- \frac{1}{7}$

ответ: $x_1=0; \ x_2=- \frac{1}{7}$

$x^3+x^2-x-1=0\\ (x-1)(x+1)^2 =0\\\\ x-1=0\\ x=1\\\\ x+1=0\\ x=-1$
ответ: $x_1=-1; \ x_2=1$

$x^3-4x^2-25x+100=0\\ (x-5)(x^2+x-20)=0\\\\ x-5=0\\ x=5\\\\ x^2+x-20=0\\ D=1+80=81; \sqrt{D}=9\\\\ x_{1/2}= \frac{-1\pm9}{2}\\\\ x_1= -\frac{10}{2}=-5\\\\ x_2= \frac{8}{2}=4$

ответ: $x_1=-5;\ x_2=4;\ x_3=5$