jmc

We rearrange the terms to see that $x^2=64$. It follows that $x=8$ or $x=-8$, so the sum of all solutions is $\boxed{0}$.

- OR -

The equation should jump out as a 6-8-10 Pythagorean triple. Thus, $x=8$ or $x=-8$, and the sum of all solutions is $\boxed{0}$, as before.