# How do you FOIL x^2+111x-104000 ?

Jan 20, 2018

${x}^{2} + 111 x - 104000$

$= \left(x + \frac{111}{2} - \frac{\sqrt{428321}}{2}\right) \left(x + \frac{111}{2} + \frac{\sqrt{428321}}{2}\right)$

#### Explanation:

I think you are wanting to factor ${x}^{2} + 111 x - 104000$

We find:

$4 \left({x}^{2} + 111 x - 104000\right)$

$= 4 {x}^{2} + 444 x - 416000$

$= {\left(2 x\right)}^{2} + 2 \left(2 x\right) \left(111\right) + {111}^{2} - 428321$

$= {\left(2 x + 111\right)}^{2} - {\left(\sqrt{428321}\right)}^{2}$

$= \left(\left(2 x + 111\right) - \sqrt{428321}\right) \left(\left(2 x + 111\right) + \sqrt{428321}\right)$

$= \left(2 x + 111 - \sqrt{428321}\right) \left(2 x + 111 + \sqrt{428321}\right)$

So:

${x}^{2} + 111 x - 104000$

$= \frac{1}{4} \left(2 x + 111 - \sqrt{428321}\right) \left(2 x + 111 + \sqrt{428321}\right)$

$= \left(x + \frac{111}{2} - \frac{\sqrt{428321}}{2}\right) \left(x + \frac{111}{2} + \frac{\sqrt{428321}}{2}\right)$

Note that $428321 = 107 \cdot 4003$ has no square factors, so $\sqrt{428321}$ is in simplest form.