Question

解方程：3x/(x²-1）+(x²-1）/x=7/2

Answer 1

设x/（x²-1)=y
则原方程为3y+1/y=7/2
即6y²-7y+2=0
(3y-2)(2y-1)=0
解得y=2/3或y=1/2
所以x/（x²-1)=2/3或x/(x²-1)=1/2
3x=2x²-2或2x=x²-1
2x²-3x-2=0或x²-2x-1=0
（2x+1)(x-2)=0或(x-1)²=2
解得x=-1/2或x=2或x=1±√2

Answer 2

令a=x/(x²-1)
则3a+1/a=7/2
6a²-7a+2=0
(2a-1)(3a-2)=0
a=1/2,a=2/3

x/(x²-1)=1/2
x²-x-1=0
x=(1±√5)/2

x/(x²-1)=2/3
2x²-3x-2=0
(2x+1)(x-2)=0
x=-1/2,x=2

所以
x=(1-√5)/2，x=(1+√5)/2，x=-1/2，x=2