  We prove that in a torsion-free hyperbolic group $\Gamma$, the length of the
value of each variable in a minimal solution of a quadratic equation $Q=1$ is
bounded by $N|Q|^3$ for an orientable equation, and by $N|Q|^{4}$ for a
non-orientable equation, where $|Q|$ is the length of the equation, and the
constant $N$ can be computed. We show that the problem, whether a quadratic
equation in $\Gamma$ has a solution, is in NP, and that there is a PSpace
algorithm for solving arbitrary equations in $\Gamma$. If additionally $\Gamma$
is non-cyclic, then this problem (of deciding existence of a solution) is
NP-complete. We also give a slightly larger bound for minimal solutions of
quadratic equations in a toral relatively hyperbolic group.
