Let $x$ and $y$ be complex numbers such that $x+y=\sqrt{20}$ and $x^{2}+y^{2}=15$. Compute $|x-y|$.
We have $(x-y)^{2}+(x+y)^{2}=2\left(x^{2}+y^{2}\right)$, so $(x-y)^{2}=10$, hence $|x-y|=\sqrt{10}$. $\fbox{\sqrt{10}}$.