1) if p = 2 then and we have a double root;

2) If p > 2 then the equation has a solution iff the quadratic's determinant is a square modulo p, i.e. is a square modulo p is a square modulo p.

For example, for p = 5,13,17,29, 97 we have a solution, pero not for p = 3, 7, 19, 31

Tonio