Existence of Solutions of a Quadratic Congruence

If a quadratic congruence modulo j;

\[n\]

has solutions, then the discriminant must be a quadratic residue ,modulo j;

\[n\]

.
Example: Does the congruence j;

\[x^2+8x+4 \equiv 0 /; (mod \; 13)\]

have solutions?
The discriminant is j;

\[8^2-4 \times 1 \times 4=48 \equiv 9=3^2 \; (mod \; 13) \]

so 9 is a quadratic residue of 13 and the congruence has solutions.