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 $x^2+bx+c\equiv0\bmod q^2$ where $b^2-4c\equiv0\bmod q$ but $b^2-4c\not\equiv0\bmod q^2$?

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.