Similar to Thomas's answer, there are problems that can be solved without complex numbers, but if you solve for $x$ it is complex.
E.g. $x^3+1/x^3=1$ evaluate $x^{999}+1/x^{999}$
(You can see $x$ is actually complex here, and WolframAlpha also shows the solution here, but I've forgotten the non-complex solution :-( )