# How do you solve x^2 + 10x + 5 = 0  by completing the square?

May 13, 2016

$x = - 5 \pm 2 \sqrt{5}$
(see below for completing the squares method of solution)

#### Explanation:

Given:
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} + 10 x + 5 = 0$

Move the constant to the right side as
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{{x}^{2} + 10 x} = - 5$

We know that ${\left(x + a\right)}^{2} = \textcolor{red}{{x}^{2} + 2 a x + {a}^{2}}$
So if the first two terms of a squared binomial are
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{{x}^{2} + 2 a x} = \textcolor{b l u e}{{x}^{2} + 10 x}$
then
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a} = \textcolor{b l u e}{5}$
and we will need to add
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{{a}^{2}} = \textcolor{b l u e}{25}$ (to both sides) to complete the square:

$\textcolor{w h i t e}{\text{XXX}} {x}^{2} + 10 x + 25 = - 5 + 25$

Writing as a squared binomial and simplifying the right side:
$\textcolor{w h i t e}{\text{XXX}} {\left(x + 5\right)}^{2} = 20$

Taking the square root of both sides:
$\textcolor{w h i t e}{\text{XXX}} x + 5 = \pm 2 \sqrt{5}$

Subtracting $5$ from both sides
$\textcolor{w h i t e}{\text{XXX}} x = - 5 \pm 2 \sqrt{5}$