Answer by Asinomás for There is only one distinct pair of numbers that multiply to a given number and sum to a given number.

If two numbers add to $a$ then you can write them as $\frac{a-x}{2}$ and $\frac{a+x}{2}$. The product is therefore $\frac{a^2-x^2}{4}$, clearly $x^2=y^2\iff x=\pm y$, but this only corresponds to swapping the two numbers.