How to solve this equation?

We can immediately see by inspection that when 2x - 3 = 0, both sides of the equation are zero.

This gives us our first soution:  2x - 3 =0  so  x = 3/2

 

Now divide through by 2x - 3 (which we can do for the other solutions where 2x - 3 is not zero).

We are left with (2x - 3)² = 4(2x - 3) - 3 

 

Let u = 2x - 3, substitute in to the above and rearrange to get:  u²  - 4u + 3 = 0

This quadratic factorises as (u - 1)(u - 3) = 0, giving u = 1 and u = 3

 

These, in turn, give:  2x - 3 = 1  means x = 2, and 2x - 3 = 3 means x = 3.