Solution of polynomial congruence

Question

We all know that If $(a,m) = 1$ , Then the linear congruence $ax\equiv b(mod\ m)$ has exactly one solution. Can we conclude something similar for quadratic congruence $ax^{2}+bx+c\equiv 0(mod\ m)$ ( i.e. a condition on $a ,b ,c$ and $m$ such that it has $2$ solutions , since they always occur in pairs )?

Answer 1

The quadratic formula works mod $m$ provided $2a$ is invertible mod $m$, that is, $\gcd(2a,m)=1$.

The discriminant $\Delta = b^2-4ac$ plays a similar role as in $\mathbb R$, that is, the equation $ax^{2}+bx+c\equiv 0 \bmod m$ has:

• no solution, if $\Delta$ is not a square mod $m$

• one solution, if $\Delta=0$

• two or more solutions, if $\Delta$ is a square mod $m$

When $m$ is prime, then there are at most two solutions. This may fail when $m$ is composite. For instance, $x^2-1=0$ has four solutions mod $8$.