Mathematica spits out 8 solutions, but they areextraordinarilylong. I'm not even going to attempt to post them here, as they are multi-line solutions. Algebraically, though, there's nothing more complicated than powers and square roots and fractions, and such. I would try to find someone who has Mathematica, and enter the following command:

Simplify[Solve[{a^2 + 2b c == x, b^2 + 2a c == y, c^2 + 2a b == z}, {a, b, c}]].

That should do it.