Let a, b, c, and n be integers, with a nonzero and n at least two. Necessary and sufficient conditions on these parameters are derived which guarantee that all solutions of the congruence

are given precisely by the solutions of

where s varies over all solutions of

Corollaries of this result for prime-power moduli are also deduced and some illustrative examples are presented.

quadratic formula, quadratic residue, quadratic non-residue, congruence modulo n, quadratic congruence.