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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 $b^2-4c\equiv0\bmod q$ and the cases are 1. $b^2-4c\equiv0\bmod q^2$ and 2. $b^2-4c\not\equiv0\bmod q^2$?

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.

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 $b^2-4c\equiv0\bmod q$ and $gcd(b,q)=1$ and the cases are

1. $b^2-4c\equiv0\bmod q^2$ (most interested) and
2. $b^2-4c\not\equiv0\bmod q^2$?

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.