# How do you solve (x-2)² + (-x+6)² = 32?

##### 1 Answer
Mar 14, 2017

$x = 4 - 2 \sqrt{3} \textcolor{w h i t e}{\text{XX")orcolor(white)("XX}} x = 4 + 2 \sqrt{3}$

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{{\left(x - 2\right)}^{2}} + \textcolor{b l u e}{{\left(- x + 6\right)}^{2}} = \textcolor{b r o w n}{32}$

Expanding the left side
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{{x}^{2} - 4 x + 4} + \textcolor{b l u e}{{x}^{2} - 12 x + 36} = \textcolor{b r o w n}{32}$

Combining like terms
$\textcolor{w h i t e}{\text{XXX}} 2 {x}^{2} - 16 x + 40 = \textcolor{b r o w n}{32}$

Dividing both sides by $2$ to simplify
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} - 8 x + 20 = 16$

Converting to standard quadratic form
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{1} {x}^{2} \textcolor{m a \ge n t a}{- 8 x} \textcolor{\mathmr{and} a n \ge}{+ 4} = 0$

With no obvious (at least to me) rational factors:
Apply the quadratic formula
$\textcolor{w h i t e}{\text{XXX}} x = \frac{- \textcolor{m a \ge n t a}{b} \pm \sqrt{{\textcolor{m a \ge n t a}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{\mathmr{and} a n \ge}{c}}}{2 \textcolor{red}{a}}$

$\textcolor{w h i t e}{\text{XXXXX}} = \frac{8 \pm \sqrt{64 - 16}}{2}$

$\textcolor{w h i t e}{\text{XXXXX}} = 4 \pm 2 \sqrt{3}$