# How do you factor 2x^2+5x-30?

May 26, 2016

$2 {x}^{2} + 5 x - 30 = \frac{1}{8} \left(4 x + 5 - \sqrt{265}\right) \left(4 x + 5 + \sqrt{265}\right)$

#### Explanation:

Complete the square, first premultiplying by $8$ to cut down on fraction arithmetic, not forgetting to divide by $8$ at the end:

$8 \left(2 {x}^{2} + 5 x - 30\right)$

$= 16 {x}^{2} + 40 x - 240$

$= {\left(4 x\right)}^{2} + 2 \left(4 x\right) \left(5\right) - 240$

$= {\left(4 x + 5\right)}^{2} - 25 - 240$

$= {\left(4 x + 5\right)}^{2} - 265$

$= {\left(4 x + 5\right)}^{2} - \left({\sqrt{265}}^{2}\right)$

$= \left(\left(4 x + 5\right) - \sqrt{265}\right) \left(\left(4 x + 5\right) + \sqrt{265}\right)$

$= \left(4 x + 5 - \sqrt{265}\right) \left(4 x + 5 + \sqrt{265}\right)$

Dividing both ends by $8$, we find:

$2 {x}^{2} + 5 x - 30 = \frac{1}{8} \left(4 x + 5 - \sqrt{265}\right) \left(4 x + 5 + \sqrt{265}\right)$