How do you solve using completing the square method x^2+5x-2=0?

1 Answer
Oct 30, 2017

$x = - \frac{5}{2} \pm \frac{\sqrt{33}}{2}$

Explanation:

The difference of square identity can be written:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Use this with $a = \left(2 x + 5\right)$ and $b = \sqrt{33}$

First premultiply by $4$ to reduce the amount of fraction arithmetic we need to do...

$0 = 4 \left({x}^{2} + 5 x - 2\right)$

$\textcolor{w h i t e}{0} = 4 {x}^{2} + 20 x - 8$

$\textcolor{w h i t e}{0} = {\left(2 x\right)}^{2} + 2 \left(2 x\right) \left(5\right) + {5}^{2} - 33$

$\textcolor{w h i t e}{0} = {\left(2 x + 5\right)}^{2} - {\left(\sqrt{33}\right)}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(2 x + 5\right) - \sqrt{33}\right) \left(\left(2 x + 5\right) + \sqrt{33}\right)$

$\textcolor{w h i t e}{0} = \left(2 x + 5 - \sqrt{33}\right) \left(2 x + 5 + \sqrt{33}\right)$

So:

$2 x = - 5 \pm \sqrt{33}$

So:

$x = - \frac{5}{2} \pm \frac{\sqrt{33}}{2}$