If $q$ is a prime what is the best method to compute roots of a quadratic polynomial $f(x)\equiv0\bmod q^2$ which is of form $ax^2+bx+c\equiv0\bmod q^2$ where $c\bmod q\equiv0$?

If $q$ is composite with prime factorization known what is the best way?

If factorization of $q$ is unknown I believe the problem is at least as hard as factoring.