# Question #2cf05

Oct 12, 2017

$y = 2 {\left(x - \left(\frac{3}{2}\right)\right)}^{2} - \left(\frac{3}{2}\right)$
$v e r t e x \left(\frac{3}{2} , - \left(\frac{3}{2}\right)\right)$

#### Explanation:

$y = 2 {x}^{2} - 6 x + 3$
$y - 3 = 2 {x}^{2} - 3 x$
$y - 3 = 2 \left({x}^{2} - \left(\frac{3}{2}\right) x\right)$
$y - 3 = 2 \left({x}^{2} - \left(\frac{3}{2}\right) x + + \frac{9}{4}\right) - \left(\frac{9}{2}\right)$
$y = 2 {\left(x - \left(\frac{3}{2}\right)\right)}^{2} - \left(\frac{9}{2}\right) + 3$
$y = 2 {\left(x - \left(\frac{3}{2}\right)\right)}^{2} - \left(\frac{3}{2}\right)$

$h = \frac{3}{2} , k = - \left(\frac{3}{2}\right)$

Oct 12, 2017

The vertex of $y = 2 {x}^{2} - 6 x + 3$ is $y = 2 {\left(x - \frac{3}{2}\right)}^{2} + \frac{3}{2}$

#### Explanation:

To solve this problem, we need to convert this quadratic equation from standard form to vertex form.

$y = 2 {x}^{2} - 6 x + 3$

Now let's turn the coefficient of the ${x}^{2}$ term into $1$. This way, we're setting the stage for the process of completing the square.

$\frac{1}{2} y = {x}^{2} - 3 x + \frac{3}{2}$

Now we can start completing the square:

$\frac{1}{2} y + {\left(- \frac{3}{2}\right)}^{2} = {x}^{2} - 3 x + {\left(- \frac{3}{2}\right)}^{2} + \frac{3}{2}$

Then we simplify the left side and write ${x}^{2} - 3 x + {\left(- \frac{3}{2}\right)}^{2}$ as the square of a binomial (a two-term polynomial):

$\frac{1}{2} y + \frac{9}{4} = {\left(x - \frac{3}{2}\right)}^{2} + \frac{3}{2}$

We then subtract $\frac{9}{4}$ from both sides and simplify:

$\frac{1}{2} y = {\left(x - \frac{3}{2}\right)}^{2} + \frac{3}{2} - \frac{9}{4} = {\left(x - \frac{3}{2}\right)}^{2} + \frac{6}{4} - \frac{9}{4} = {\left(x - \frac{3}{2}\right)}^{2} - \frac{3}{4}$

All we need to do now is multiply by $2$ to isolate $y$ and obtain our vertex form:

$y = 2 \left({\left(x - \frac{3}{2}\right)}^{2} - \frac{3}{4}\right) = 2 {\left(x - \frac{3}{2}\right)}^{2} - 2 \left(- \frac{3}{4}\right) = 2 {\left(x - \frac{3}{2}\right)}^{2} + \frac{3}{2}$

And so our vertex form is:

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