# What is the vertex form of y=(6x + 3)(x - 5) ?

May 30, 2017

$6 {\left(x - \frac{9}{4}\right)}^{2} - \frac{363}{8}$

#### Explanation:

The vertex form of a quadratic equation is $a {\left(x - h\right)}^{2} + k$.

We have: $y = \left(6 x + 3\right) \left(x - 5\right)$

To express this equation in its vertex form, we must "complete the square".

First, let's expand the parentheses:

$R i g h t a r r o w y = 6 {x}^{2} - 30 x + 3 x - 15$

$R i g h t a r r o w y = 6 {x}^{2} - 27 x - 15$

Then, let's factor $6$ out of the equation:

$R i g h t a r r o w y = 6 \left({x}^{2} - \frac{27}{6} x - \frac{15}{6}\right)$

$R i g h t a r r o w y = 6 \left({x}^{2} - \frac{9}{2} x - \frac{5}{2}\right)$

Now, let's add and subtract the square of half of the $x$ term within the parentheses:

$R i g h t a r r o w y = 6 \left({x}^{2} - \frac{9}{2} x + {\left(\frac{9}{4}\right)}^{2} - \frac{5}{2} - {\left(\frac{9}{4}\right)}^{2}\right)$

$R i g h t a r r o w y = 6 \left({\left(x - \frac{9}{4}\right)}^{2} - \frac{5}{2} - \frac{81}{16}\right)$

$R i g h t a r r o w y = 6 \left({\left(x - \frac{9}{4}\right)}^{2} - \frac{121}{16}\right)$

Finally, let's distribute $6$ throughout the parentheses:

$\therefore = 6 {\left(x - \frac{9}{4}\right)}^{2} - \frac{363}{8}$