Question

`( (x - 1)/(x + 2) )^(2) - (3)/(2) = (x - 1)/(2x + 4)`

Can you solve for x​

Answer 1

Yes I can. But that's probably not what you are asking. So I'll just assume you wan't me to figure out the value x.

We have the following equality

`((x-1)/(x+2))^2-(3)/(2)=(x-1)/(2x+4)`.

Let's rearrange it so that we have terms with x on the left side and 3/2 on the right side

`((x-1)/(x+2))^2-(x-1)/(2x-4)=(3)/(2)`

We now use this rule
`((a)/(b))^n=(a^n)/(b^n)` to epand the first fraction. In second fraction we just factor 2 out of the denominator

`(x^2-2x+1)/((x+2)^2)-(x-1)/(2(x+2))=(3)/(2)`.

Simplify further to get

`(2(x^2-2x+1)-((x+2)(x-1)))/((x+2)^2)=(3)/(2)\\(2x^2-4x+2-(x^2+x-2))/((x+2)^2)=(3)/(2)\\(2x^2-4x+2-x^2-x+2)/((x+2)^2)=(3)/(2)\\(x^2-5x+4)/((x+2)^2)=(3)/(2)`

Cross multiply to get

`2(x^2-5x+4)=3((x+2)^2)`

Simplify even more

`2x^2-10x+8=3x^2+12x+12\\\ -x^2-22x-4=0\\\ x^2+22x+4=0`

Now using quadratic formula we get two solutions

`x_1 =\boxed{-11+3√(13)}, x_2=\boxed{-11-3√(13)}`

Hope this helps.