# How do you solve x^2 + 32x + 15 = -x^2 + 16x + 11 by completing the square?

Jun 22, 2015

Simplify the given equation then apply the process of "completing the squares" to obtain
$\textcolor{w h i t e}{\text{XXXX}}$$x = - 4 \pm \sqrt{14}$

#### Explanation:

Given ${x}^{2} + 32 x + 15 = - {x}^{2} + 16 x + 11$

First simplify to get all terms involving $x$ on the left side and a simple constant on the right:
$\textcolor{w h i t e}{\text{XXXX}}$$2 {x}^{2} + 16 x = - 4$
Further simplify by dividing by 2
$\textcolor{w h i t e}{\text{XXXX}}$${x}^{2} + 8 x = - 2$

Now we are ready to begin completing the square.
#

Noting that the squared binomial ${\left(x + a\right)}^{2} = {x}^{2} + 2 a x + {a}^{2}$

If ${x}^{2} + 8 x$ are the first 2 terms of an expanded squared binomial,
then $2 a x = 8 x \rightarrow a = 4 \mathmr{and} {a}^{2} = 16$
So the completed square must be (after remembering that anything added to one side of an equation must also be added to the other)
$\textcolor{w h i t e}{\text{XXXX}}$${x}^{2} + 8 x + 16 = - 2 + 16$
or
$\textcolor{w h i t e}{\text{XXXX}}$${\left(x + 4\right)}^{2} = 14$

Taking the square root of both sides:
$\textcolor{w h i t e}{\text{XXXX}}$$x + 4 = \pm \sqrt{14}$
and
$\textcolor{w h i t e}{\text{XXXX}}$$x = - 4 \pm \sqrt{14}$