jmc

Knowing that $117=9\times 13$, we immediately reduce the problem to finding the sum of all possible solutions of $81x^2+169-250=0$. Seeing that this is a quadratic whose linear coefficient is 0, the solutions sum to $\boxed{0}$.

To find the solutions, we can rewrite $81x^2-81=0⟹x^2=1$, so the solutions are $1,-1$.