# How do you factor n^2+8n=12?

May 28, 2015

To be technically accurate (at least as I was taught) you factor expressions not equations.
I will assume that what you really want to do is find solutions for the given equation ${n}^{2} + 8 n = 12$
(and then re-write the equation as a pair of factors equal to zero).

There are several ways to approach this:
trial and error;
completion of the square.

By completion of the square
${n}^{2} + 8 n = 12$

$\rightarrow {n}^{2} + 8 n + {4}^{2} = 12 + {4}^{2}$

$\rightarrow {\left(n + 4\right)}^{2} = 28$

$\rightarrow \left(n + 4\right) = \pm \sqrt{28}$

$\rightarrow n = - 4 \pm 2 \sqrt{7}$

Re-expressing the original equation as a pair of factors equal to zero
$\left(n + 4 + 2 \sqrt{7}\right) \left(n + 4 - 2 \sqrt{7}\right) = 0$