解方程:x^2-6x-6-x√(x^2-2x-2)=0解方程：x^2-6x-6-x√(x^2-2x-2)=0

Question

解方程:x^2-6x-6-x√(x^2-2x-2)=0
解方程：x^2-6x-6-x√(x^2-2x-2)=0

Answer 1

x^2-6x-6-x√(x^2-2x-2)=0
显然x不等于0,两边同时除以x^2,得
1-6（x+1/x^2)-√[1-2(x+1/x^2)]=0
令 y=√[1-2(x+1/x^2)] (y>0)
则 y^2=1-2(x+1/x^2) .(1)
原方程变为 3y^2-y-2=0
y=1,y=-2/3(不合要求,舍去)
将y=1代入（1）得
x+1/x^2=0
=>x^3+1=0
=>(x+1)(x^2-x+1)=0
实数解为 x=-1
验证 x=-1 是原方程的解.