I am familiar with using the quadratic formula and Tonelli-Shanks with Hensel's Lifting Lemma to solve a quadratic equation, but how do I solve a quadratic equation in an even modulus? I can't use the quadratic formula because of the division by two. Would it be sufficient to multiply both sides of the equation (including modulus) by 2?
In particular, if I try to solve:
$x^2 + bx + c\equiv 0 \ (mod\ 2pq)$, where $p\equiv q\equiv 1\ (mod\ 2)$
I would end up with:
$\frac{-b \pm \sqrt{b^2-4c}}{2} \equiv 0\ (mod\ 2pq)$
$= -b \pm \sqrt{b^2-4c} \equiv 0\ (mod\ 4pq)$
does this work? Or am I terribly, terribly wrong?
edit: To start with a simple one: $x^2 + 5x + 6 \equiv 0\ (mod\ 28)$ the solutions are: $\{5,18,25,26\}$ I was able to find them using the quadratic formula without the division by 2 and then testing both possible solutions to the multiplicative inverse of 2 from the results there. However, I wasn't able to determine a pattern between finding those solutions and testing them versus checking the equation mod 56 either.
edit: corrected the solution set for the example.