Solve the equation `x^(2)+(x/(x-1))^(2)=8`

Answer

`x_(1)=2,x_(2)=-1+sqrt(3)` and `x_(3)=-1-sqrt(3)`

Explanation

We have `x^(2)+(x/(x-1))^(2)=8`
`implies(x+x/(x-1))^(2)-2.x.x/((x-1))=8`
`implies(x^(2))/(x-1))^(2)-2((x^(2))/(x-1))-8=0`.........i
Let `y=(x^(2))/(x-1)`. Then Eq. (i) reduces to
`y^(2)-2y-8=0`
`implies(y-4)(y+2)=0`
`:.y=4,-2`
If `y=4` then `4=(x^(2))/(x-1)`
or `x^(2)-4x+4=0`
or `(x-2)^(2)=0`
or `x=2`
`:.x_(1)=2`
and if `y=-2`, then `-2=(x^(2))/(x-1)`
or `x^(2)+2x-2=0`
`:.x=(-2+-sqrt((4+8)))/2`
`impliesx=-1+-sqrt(3)`
`:.x_(2)=-1+sqrt(3),x_(3)=-1-sqrt(3)`