换元法解方程x^2-x+1=6/x^2-x
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换元法解方程x^2-x+1=6/x^2-x
换元法解方程x^2-x+1=6/x^2-x
换元法解方程x^2-x+1=6/x^2-x
令a=x^2-x
则a+1=6/a
a^2+a=6
a^2+a-6=0
(a+3)(a-2)=0
a=-3,a=2
x^2-x=a=-3
x^2-x+3=0
判别式b^2-4ac<0,无解
x^2-x=a=2
x^2-x-2=0
(x-2)(x+1)=0
所以x=2,x=-1
X=2或-1
x^2-x+1=6/x^2-x
令m=x^2-x
则原方程变为:
m+1=6/m
m^2+m-6=0
(m-2)(m+3)=0
m=2或m=-3
x^2-x=2
x^2-x-2=0
(x+1)(x-2)=0
x=-1或x=2
x^2-x=-3
x^2-x+3=0
△=1-12=-11
所以方程无解
所以原方程的解为:
x=-1或x=2