Question

Solve for x
(X-P)^2 =(X- q)^2

Answer 1

(x-p)^2 = (x-q)^2
just take the root, and you have either
x-p = x-q
so no solution unless p=q, in which case there are infinitely many solutions
OR
x-p = -(x-q)
2x = q+p
x = (q+p)/2

Or, you could have expanded at the start
(x-p)^2 = (x-q)^2
x^2 - 2px + p^2 = x^2 - 2qx + q^2
2(q-p)x = q^2 - p^2 = (q-p)(q+p)
2x = q+p unless q=p
x = (q+p)/2