Question

Say you have a quadratic congruence equation. Something like ax^2+bx+c \equiv 0 (mod p), and then you try to solve it by completing the square into something like (cx+d)^2 \equiv f (mod p), ie, you now have a problem like y^2 \equiv f (mod p) where y = cx+d). Say you determine that there is a solution to that problem... how do you know that the solution is of the form y=cx+d? If it's not of that form then it doesn't satisfy the original congruence equation.