# What is the vertex form of y=-3x^2 - 5x + 9 ?

Apr 11, 2018

$y = - 3 {\left(x + \frac{5}{6}\right)}^{2} + \frac{133}{12}$

#### Explanation:

$y = - 3 \left[{x}^{2} + \frac{5}{3}\right] + 9$
$y = - 3 \left[{\left(x + \frac{5}{6}\right)}^{2} - \frac{25}{36}\right] + 9$
$y = - 3 {\left(x + \frac{5}{6}\right)}^{2} + \frac{25}{12} + 9$
$y = - 3 {\left(x + \frac{5}{6}\right)}^{2} + \frac{133}{12}$

Apr 11, 2018

Given: $y = - 3 {x}^{2} - 5 x + 9$

Write as: $y = - 3 \left({x}^{2} + \frac{5}{3} x\right) + 9 \text{ } \ldots \ldots \ldots \ldots \ldots . . E q u a t i o n \left(1\right)$

Consider the $\left(\textcolor{g r e e n}{{x}^{2} + \frac{5}{3} x}\right)$ part

We need to make this a $\underline{\text{'perfect square'}}$ but in 'forcing' it to do this we introduce a value that is not in the original equation. To correct this we have to turn it into 0 by subtraction or addition as appropriate by the same amount. Rather like $a + 2$ being changed to $\left(a + 2\right) + 3 - 3$

color(green)(-3[x^2+5/3x] color(white)("ddd")->color(white)("ddd")-3[(x+5/(2xx3))^2]

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddddddd")->color(white)("ddd}} - 3 \left[{x}^{2} + \frac{5}{3} x \textcolor{red}{\textcolor{w h i t e}{.} \underbrace{+ {\left(\frac{5}{6}\right)}^{2}}}\right]}$
$\textcolor{w h i t e}{\text{ddddddddddddddddddddddddddddddddd.d}} \textcolor{red}{\uparrow}$
$\textcolor{w h i t e}{\text{dddddddddddddddddddddddd")color(red)("The introduced error}}$

Substitute this into $E q u a t i o n \left(1\right)$

color(green)(y=-3(x^2+5/3x)+9

$\textcolor{w h i t e}{\text{dddddddddddddddd")color(red)("The error}}$
$\textcolor{w h i t e}{\text{ddddddddddddddddd.d}} \textcolor{red}{\downarrow}$
$\textcolor{g r e e n}{y = \underbrace{- 3 \left[{x}^{2} + \frac{5}{3} x \textcolor{red}{\textcolor{w h i t e}{.} + \overbrace{{\left(\frac{5}{6}\right)}^{2}}}\right]} + \textcolor{b l u e}{k} + 9} \text{ } k$ is the correction
$\textcolor{w h i t e}{\text{ddddddddddd.d}} \textcolor{g r e e n}{\downarrow}$
color(green)(y=color(white)("ddd")-3(x+5/6)^2color(white)("ddddd")+color(blue)(k)+9

The whole error is $\textcolor{red}{\left(- 3\right) \times {\left(\frac{5}{6}\right)}^{2}}$

$\textcolor{g r e e n}{y = \textcolor{w h i t e}{\text{ddd}} - 3 {\left(x + \frac{5}{6}\right)}^{2} + \textcolor{b l u e}{\left[3 \times {\left(\frac{5}{6}\right)}^{2}\right]} + 9}$

$\textcolor{w h i t e}{}$

$y = - 3 {\left(x + \frac{5}{6}\right)}^{2} + \frac{133}{12}$