What is the vertex form of y=3x^2-39x-90 ?

May 31, 2018

$y = 3 {\left(x - \frac{13}{2}\right)}^{2} - \frac{867}{4}$
$\textcolor{w h i t e}{\text{XXX}}$ with vertex at $\left(\frac{13}{2} , - \frac{867}{4}\right)$

Explanation:

The general vertex form is $y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$ with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$

Given:
$y = 3 {x}^{2} - 39 x - 90$

extract the dispersion factor ($\textcolor{g r e e n}{m}$)
$y = \textcolor{g r e e n}{3} \left({x}^{2} - 13 x\right) - 90$

complete the square
$y = \textcolor{g r e e n}{3} \left({x}^{2} - 13 x \textcolor{m a \ge n t a}{+ {\left(\frac{13}{2}\right)}^{2}}\right) - 90 \textcolor{m a \ge n t a}{- \textcolor{g r e e n}{3} \cdot {\left(\frac{13}{2}\right)}^{2}}$

re-writing the first term as a constant times a squared binomial
and evaluating $- 90 - 3 \cdot {\left(\frac{13}{2}\right)}^{2}$ as $- \frac{867}{4}$

y=color(green)3(x-color(red)(13/2))^2+color(blue)(""(-867/4))

May 31, 2018

Vertex form of equation is $y = 3 {\left(x - 6.5\right)}^{2} - 216.75$

Explanation:

$y = 3 {x}^{2} - 39 x - 90$ or

$y = 3 \left({x}^{2} - 13 x\right) - 90$ or

$y = 3 \left({x}^{2} - 13 x + {6.5}^{2}\right) - 3 \cdot {6.5}^{2} - 90$ or

$y = 3 {\left(x - 6.5\right)}^{2} - 126.75 - 90$ or

$y = 3 {\left(x - 6.5\right)}^{2} - 216.75$

Vertex is $6.5 , - 216.75$ and

Vertex form of equation is $y = 3 {\left(x - 6.5\right)}^{2} - 216.75$

graph{3x^2-39x-90 [-640, 640, -320, 320]} [Ans]