Let \(u\) and \(v\) be real numbers such that \(u^3 - 3uv^2 = 259, 3u^2 v - v^3 = -286\). Find \(u^2 + v^2\).

\(u^3 -3uv^2 + 3u^2 v - v^3 = (u-v)(u^2 + 4uv+v^2) = 259-286= -27\)

\(\text{This has to be factored as }\\ (u-v)=9\\ (u^2+4uv+v^2)=-3\\ \text{in order to match the equation, and further must be that }\\ u=7, v=-2\\ u^2+v^2 = 49+4 = 53\)