Existence of Solutions of a Quadratic Congruence
\[n\]has solutions, then the discriminant must be a quadratic residue ,modulo  j;
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.