# What is the vertex form of y=13x^2 +3x- 36 ?

Feb 1, 2016

vertex form: $y = {\left(x + \frac{3}{26}\right)}^{2} - \frac{1881}{52}$

#### Explanation:

1. Factor 13 from the first two terms.

$y = 13 {x}^{2} + 3 x - 36$

$y = 13 \left({x}^{2} + \frac{3}{13} x\right) - 36$

2. Turn the bracketed terms into a perfect square trinomial.
When a perfect square trinomial is in the form $a {x}^{2} + b x + c$, the $c$ value is ${\left(\frac{b}{2}\right)}^{2}$. Thus you divide $\frac{3}{13}$ by $2$ and square the value.

$y = 13 \left({x}^{2} + \frac{3}{13} x + {\left(\frac{3}{13} x \div 2\right)}^{2}\right) - 36$

$y = 13 \left({x}^{2} + \frac{3}{13} x + \frac{9}{676}\right) - 36$

3. Subtract 9/676 from the perfect square trinomial.
You cannot just add $\frac{9}{676}$ to the equation, so you must subtract it from the $\frac{9}{676}$ you just added.

y=13(x^2+3/13x+9/676 color(red)(-9/676))-36

4. Multiply -9/676 by 13.
The next step is to bring $- \frac{9}{676}$ out of the brackets. To do this, multiply $- \frac{9}{676}$ by the $a$ value, $13$.

$y = \textcolor{b l u e}{13} \left({x}^{2} + \frac{3}{13} x + \frac{9}{676}\right) - 36 \left[\textcolor{red}{\left(- \frac{9}{676}\right)} \cdot \textcolor{b l u e}{\left(13\right)}\right]$

5. Simplify.

$y = \left({x}^{2} + \frac{3}{13} x + \frac{9}{676}\right) - 36 - \frac{9}{52}$

$y = \left({x}^{2} + \frac{3}{13} x + \frac{9}{676}\right) - \frac{1881}{52}$

6. Factor the perfect square trinomial.
The last step is to factor the perfect square trinomial. This will allow you to determine the coordinates of the vertex.

$\textcolor{g r e e n}{y = {\left(x + \frac{3}{26}\right)}^{2} - \frac{1881}{52}}$

$\therefore$, the vertex form is $y = {\left(x + \frac{3}{26}\right)}^{2} - \frac{1881}{52}$.