Finding 2 numbers from their product and sum

Given two real numbers $r$ and $s$ , if $u$ is their sum and $p$ their product, $r$ and $s$ are:

$\frac{u}{2} \pm \frac{\sqrt{u^2-4p}}{2}$

This is basically the quadratic formula – note that $(x-r)(x-s)=x^2-(r+s)x+rs$ .

Derivation using quadratic formula

$u=r+s$ and $p=rs$ , so:

$\begin{align*} r&=p/s=u-s\\ p&=s(u-s)=su-s^2\\ s^2-su+p&=0\\ s&=\frac{u\pm\sqrt{u^2-4p}}{2} \end{align*}$

Since $r$ and $s$ are symmetrical in $u$ and $p$ , the same equation gives $r$ .

Graphical derivation

Rendered by QuickLaTeX.com

The difference in area between the red square $(r-s)^2$ and the largest square $u^2=(r+s)^2$ is evidently equal to 4 of the green rectangles each of area $rs=p$ . (See in the right picture that the dark green overlap $s^2$ is equal in area to the square at top right not covered.)

$\begin{align*} r&=u-s\\ r-s&=u-2s\\ (r-s)^2&=(u-2s)^2 \\ &=(2s-u)^2\\ &=u^2-4p \quad \text{as shown above}\\ \text{So } 2s-u&=\pm\sqrt{u^2-4p}\\ s&=\frac{u\pm\sqrt{u^2-4p}}{2}\\ \end{align*}$

Since similarly $s-r=u-2r$ , and $(r-s)^2=(s-r)^2$ , the RHS also gives $r$ .

Pic from E. Hairer and G. Wanner, Analysis by its History, 2008.

Graph of $x+y$ and $xy$ :

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