Quadratic congruence ($a{x}^2+bx+c\equiv 0(\mathrm{mod}n)$)

A quadratic congruence is a modular congruence defined by a degree-two polynomial, with solutions sought among residue classes. For odd $n$ and $a$ invertible modulo $n$, completing the square gives $(2ax+b{)}^2\equiv b^2-4ac(\mathrm{mod}n)$. A Jacobi symbol of $-1$ for a coprime proposed square rules out solutions, whereas a Jacobi symbol of $1$ alone does not establish solubility for a composite modulus.