Question

用换元法解方程

用换元法解方程x分之(x-1)-(x-1)分之3x+1=0如果设x分之x-1=y 将原方程化为关于y的整式方程 ！！！！！ 快教我

Answer 1

x^2-x-12/(x^2-x)=4
令a=x^2-x
a-12/a=4
a^2-12=4a
a^2-4a-12=0
(a-6)(a+2)=0
a=6或a=-2

x^2-x=6
x^2-x-6=0
(x-3)(x+2)=0
x=3或x=-2

x^2-x=-2
x^2-x+2=0
△=1^2-4*1*2
=-7<0
所以方程无解

经检验x=3或x=-2是方程的解

3x^2/(x+1)-(x+1)/x^2=2
令x^2/(x+1)=m
3m-1/m=2
3m^2-1=2m
3m^2-2m-1=0
(3m+1)(m-1)=0
m=-1/3或m=1

x^2/(x+1)=-1/3
3x^2=-x-1
3x^2+x+1=0
△=1^2-4*3*1
=-11<0
所以方程无解

x^2/(x+1)=1
x^2=x+1
x^2-x-1=0
x^2-x+1/4-1/4-1=0
(x-1/2)^2-5/4=0
(x-1/2-√5/2)(x-1/2+√5/2)=0
x=1/2+√5/2或x=1/2-√5/2
经检验x=1/2+√5/2或x=1/2-√5/2是方程的解

△=1^2-4*3*1
=-11<0
所以方程无解

Answer 2

设x分之x-1=y，则：

y-3/y+1=0

Answer 3

(X-1)/X=Y
xy=x-1
x=1/(1-y)
y-((3/(1-y)+1)/(1/(1-y)-1)
=y- (3+1-y)/(y-1+1)
=y-(2-y)/y
=0
y=(2-y)/y
y^2=2-y