# How do you convert vertex form to factored form y = 3(x+7)^2 - 2?

May 1, 2015

Expand the vertex form into standard quadratic form; then use the quadratic root formula to determine the roots.

$y = 3 {\left(x + 7\right)}^{2} - 2$

$= 3 \left({x}^{2} + 14 x + 49\right) - 2$

$= 3 {x}^{2} + 42 x + 145$

Using the formula for determining roots (and a very sharp pencil)
$\frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

gives roots at
$x = - 7 + \frac{\sqrt{6}}{3}$
and
$x = - 7 - \frac{\sqrt{6}}{3}$

So $x + 7 - \frac{\sqrt{6}}{3}$
and
$x + 7 + \frac{\sqrt{6}}{3}$
are factors of the original equation

Fully factored form
$y = 3 {\left(x + 7\right)}^{2} - 2$

$= 3 \left(x + 7 - \frac{\sqrt{6}}{3}\right) \left(x + 7 + \frac{\sqrt{6}}{3}\right)$