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Problem 322

AMC 12 late, AIME early

Algebra Difficulty 4.6 Find the answer

Find two numbers whose product and sum are given numbers. Under what conditions do we get positive rational solutions?

Your answer

Official solution

$\begin{gathered} x y=a, \\ x+y=b \\ x=\frac{b}{2} \pm \sqrt{\frac{b^{2}}{4}-a} \end{gathered}$

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

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.