Find all pairs of integers $(x, y)$ for which $x^{2}+x y=y^{2}$.
The only such pair is $(0,0)$. If $x=0$, we easily get $y=0$ which satisfies the equation. Otherwise dividing through by $x^{2}$ we get

$$
1+\frac{y}{x}=\left(\frac{y}{x}\right)^{2} .
$$

This implies $\left(\frac{y}{x}\right)^{2}-\frac{y}{x}-1=0$ and the quadratic formula gives us $\frac{y}{x}=\frac{1 \pm \sqrt{5}}{2}$. Hence $\frac{2 y}{x}=1 \pm \sqrt{5}$ or equivalently $\frac{2 y-x}{x}= \pm \sqrt{5}$. Since $\sqrt{5}$ is irrational, this is a contradiction.