I realize that b^2 - 4ac is taken from the quadratic equation, but this is a congruence -- it works well with addtion and multiplication, but division and taking the square root can't be done, and cancellation of a

common factor on both sides of a congruence can only be done

in proper circumstances.

EVERYTHING in this world can be done ONLY in proper circumstances... What your congruence modulo p says is that the quadratic equals zero in and thus you can freely use the given formula. And all the arithmetic operations , up to and including roots, can be done modulo p "under the proper circumstances" (for example, never dividing by zero...) Tonio
Right? (This is something I got from my notes, my textbook and from what my prof said.)

Thanks