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} =4in [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 0Rightarrow 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 0Rightarrow 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: $x_1= frac{5+sqrt{93}}{2}; x_2= frac{5-sqrt{93}}{2}$