化简和解分式方程

题目

化简和解分式方程
化简(x+2)/(x^2-2x) - (x-1)/(x^2-4x+4) /(1-(4/x)
解分式方程x/(x-4)=(2x+2)/(x^2-16) +1

答案

(x+2)/(x^2-2x) - (x-1)/(x^2-4x+4) /(1-(4/x)
=[(x+2)/x(x-2)-(x-1)/(x-2)^2]/[(x-4)/x]
={[(x-2)(x+2)-x(x-1)]/x(x-2)^2}/[(x-4)/x]
=[(x^2-4-x^2+x)/x(x-2)^2}/[(x-4)/x]
=(x-4)/x(x-2)^2}/[(x-4)/x]
=x(x-4)/[x(x-2)^2(x-4)]
=1/(x-2)^2
x/(x-4)=(2x+2)/(x^2-16) +1
x/(x-4)=(2x+2)/(x+4)(x-4)+1
两边乘以(x+4)(x-4)
x(x+4)=2x+2+(x+4)(x-4)
x^2+4x=2x+2+x^2-16
2x=-14
x=-7
经检验,x=-7是方程的解