Question

Решите три уравнения.ДАЮ 100 БАЛЛОВ

Answer 1

$1)(x^2-2x-1)^2+3(x-1)^2=16 \\(x^2-2x+1-2)^2+3(x-1)^2=16 \\((x-1)^2-2)^2+3(x-1)^2=16 \\(x-1)^2=y,\ y \in [0;+\infty) \\(y-2)^2+3y-16=0 \\y^2-4y+4+3y-16=0 \\y^2-y-12=0 \\D=1+48=49=7^2 \\y_1= \frac{1+7}{2} =4\in [0;+\infty) \\y_2= \frac{1-7}{2} \notin [0;+\infty) \\(x-1)^2=4 \\(x-1)^2-2^2=0 \\(x-1-2)(x-1+2)=0 \\(x-3)(x+1)=0 \\x_1=3 \\x_2=-1$
Ответ: x1=3; x2=-1
$2)(x-2)(x+1)(x+4)(x+7)=63 \\((x-2)(x+7))*((x+1)(x+4))=63 \\(x^2+7x-2x-14)(x^2+4x+x+4)=63 \\(x^2+5x-14)(x^2+5x+4)=63 \\x^2+5x=y \\(y-14)(y+4)=63 \\y^2+4y-14y-56-63=0 \\y^2-10y-119=0 \\D=100+476=576=24^2 \\y_1= \frac{10+24}{2} =17 \\y_2= \frac{10-24}{2} =-7 \\x^2+5x=-7 \\x^2+5x+7=0 \\D\ \textless \ 0\Rightarrow x \in \varnothing \\x^2+5x=17 \\x^2+5x-17=0 \\D=25+68=93 \\x_1= \frac{-5+\sqrt{93}}{2} \\x_2= \frac{-5-\sqrt{93}}{2}$
Ответ: $x_1= \frac{-5+\sqrt{93}}{2};\ x_2= \frac{-5-\sqrt{93}}{2}$
$3)(x^2-2x-8)(x^2-8x+7)=63 \\x^2-2x-8=0 \\D=4+32=36=6^2 \\x_1= \frac{2+6}{2} =4 \\x_2= \frac{2-6}{2} =-2 \\x^2-2x-8=(x-4)(x+2) \\x^2-8x+7=0 \\D=64-28=36=6^2 \\x_1= \frac{8+6}{2} =7 \\x_2= \frac{8-6}{2} =1 \\x^2-8x+7=(x-7)(x-1) \\(x^2-2x-8)(x^2-8x+7)=63 \\(x-4)(x+2)(x-7)(x-1)=63 \\((x+2)(x-7))*((x-4)(x-1))=63 \\(x^2-7x+2x-14)(x^2-x-4x+4)=63$
$(x^2-5x-14)(x^2-5x+4)=63 \\y=x^2-5x \\(y-14)(y+4)=63 \\y^2+4y-14y-56-63=0 \\y^2-10y-119=0 \\D=100+476=576=24^2 \\y_1= \frac{10+24}{2} =17 \\y_2= \frac{10-24}{2} =-7 \\x^2-5x=-7 \\x^2-5x+7=0 \\D\ \textless \ 0\Rightarrow x \in \varnothing \\x^2-5x=17 \\x^2-5x-17=0 \\D=25+68=93 \\x_1= \frac{5+\sqrt{93}}{2} \\x_2= \frac{5-\sqrt{93}}{2}$
Ответ: $x_1= \frac{5+\sqrt{93}}{2};\ x_2= \frac{5-\sqrt{93}}{2}$