# How do you solve h²=32-4h?

Mar 13, 2016

Convert to standard form then either factor or use the quadratic formula to get
$\textcolor{w h i t e}{\text{XXX}} h = - 8 \mathmr{and} h = 4$

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} {h}^{2} = 32 - 4 h$

Re-write into standard form:
color(white)("XXX")color(red)((1)h^2color(blue)(+4)hcolor(green)(-32) = 0

Option 1
Recognize the factoring:
$\textcolor{w h i t e}{\text{XXX}} \left(h + 8\right) \left(h - 4\right) = 0$
$\rightarrow \textcolor{w h i t e}{\text{XXX}} h = - 8 \mathmr{and} h = 4$

Option 2
Apply the quadratic formula for roots:
$\textcolor{w h i t e}{\text{XXX}} h = \frac{- \textcolor{b l u e}{b} \pm \sqrt{{\textcolor{b l u e}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{g r e e n}{c}}}{2 \textcolor{red}{a}}$
in this specific case:
$\textcolor{w h i t e}{\text{XXX}} h = \frac{- \textcolor{b l u e}{4} \pm \sqrt{{\left(\textcolor{b l u e}{4}\right)}^{2} - 4 \left(\textcolor{red}{1}\right) \left(\textcolor{g r e e n}{- 32}\right)}}{2 \left(\textcolor{red}{1}\right)}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{- 4 \pm \sqrt{16 + 128}}{2}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{- 4 \pm \sqrt{144}}{2}$

$\textcolor{w h i t e}{\text{XXX}} = - 2 \pm 6$

$\rightarrow \textcolor{w h i t e}{\text{XXX}} h = + 4 \mathmr{and} h = - 8$

Mar 13, 2016

$h = 4 , - 8$

#### Explanation:

${h}^{2} = 32 - 4 h$

Gather all terms to one side of the equation and arrange the equation in standard form.

${h}^{2} + 4 h - 32 = 0$

Determine two numbers that when added equal $4$ and when multiplied equal $- 32$. The numbers $8$ and $- 4$.

Rewrite the equation in factored form.

$\textcolor{red}{\left(h - 4\right)} \textcolor{b l u e}{\left(h + 8\right)} = 0$

Set each binomial equal to zero and solve for $h$.

$\textcolor{red}{h - 4} = 0$

$\textcolor{red}{h = 4}$

$\textcolor{b l u e}{h + 8} = 0$

$\textcolor{b l u e}{h = - 8}$

$h = \textcolor{red}{4} , \textcolor{b l u e}{- 8}$