Find two numbers whose product and sum are given numbers. Under what conditions do we get positive rational solutions?
$$
\begin{gathered}
x y=a, \\
x+y=b \\
x=\frac{b}{2} \pm \sqrt{\frac{b^{2}}{4}-a}
\end{gathered}
$$

The solutions are rational if $\frac{b^{2}}{4}-a$ is a positive perfect square.