# How do you write the equation in vertex form y=1/2x^2+12x-8?

Nov 17, 2016

The vertex form of this quadratic equation is $y = \frac{1}{2} {\left(x + 12\right)}^{2} - 80$.

#### Explanation:

The general vertex form of a quadratic equation is $y = a {\left(x - h\right)}^{2} + k$,
where the vertex of the graph of the function is the point $\left(h , k\right)$. To convert this quadratic equation from standard form to vertex form, we will follow the process of completing the square. We will first isolate the $x$-terms on the right side of the equation, and then factor out the coefficient of the ${x}^{2}$-term.

$y = \frac{1}{2} {x}^{2} + 12 x - 8$
$y + 8 = \frac{1}{2} {x}^{2} + 12 x - 8 + 8$

$y + 8 = \frac{1}{2} {x}^{2} + 12 x$

$y + 8 = \frac{1}{2} \left({x}^{2} + 24 x\right)$

The equation is now in the form necessary to complete the square. We will find the square of half the coefficient of the $x$-term to complete the square.

${\left(\frac{24}{2}\right)}^{2} = {12}^{2} = 144$

So, we will use $144$ to complete the square.

$y + 8 + 72 = \frac{1}{2} \left({x}^{2} + 24 x + 144\right)$

Note that we added $72$ to the left side of the equation because on the right side, the $144$ is inside parenthesis to be multiplied by $\frac{1}{2}$, and $\frac{1}{2} \left(144\right) = 72$. Now factor the right side of the equation.

$y + 80 = \frac{1}{2} {\left(x + 12\right)}^{2}$

Now, isolate the $y$-term on the left side of the equation to have the equation in vertex form.

$y + 80 - 80 = \frac{1}{2} {\left(x + 12\right)}^{2} - 80$

$y = \frac{1}{2} {\left(x + 12\right)}^{2} - 80$