# What is the vertex form of 3y=-3x^2 - 7x -2?

Nov 29, 2015

$\textcolor{g r e e n}{y = {\left(x - \frac{7}{6}\right)}^{2} - \frac{73}{36}}$
Notice I have kept it in fractional form. This is to maintain precision.

#### Explanation:

Divide through out by 3 giving:
$y = {x}^{2} - \frac{7}{3} x - \frac{2}{3}$

British name for this is: completing the square

You transform this into a perfect square with inbuilt correction as follows:

$\textcolor{b r o w n}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$
$\textcolor{b r o w n}{\text{Consider the part that is: } {x}^{2} - \frac{7}{3} x}$
$\textcolor{b r o w n}{\text{Take the"(-7/3)"and halve it. So we have} \frac{1}{2} \times \left(- \frac{7}{3}\right) = \left(- \frac{7}{6}\right)}$
$\textcolor{b r o w n}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$
Now write: $y \to {\left(x - \frac{7}{6}\right)}^{2} - \frac{2}{3}$

I have not used the equals sign because an error has been introduced. Once that error is removed we can then start to use the = sign again.
$\textcolor{w h i t e}{\times \times \times \times} \text{----------------------------------------------}$

$\textcolor{red}{\underline{\text{Finding the introduced error}}}$
If we expand the brackets we get:
color(brown)(y->x^2- 7/3 xcolor(blue)(+(7/6)^2)-2/3

The blue is the error.
$\textcolor{w h i t e}{\times \times \times \times} \text{----------------------------------------------}$

$\textcolor{red}{\underline{\text{Correction for the introduced error}}}$
We correct for this by subtracting the same value so that we have:

color(brown)(y->x^2- 7/3 xcolor(blue)(+(7/6)^2-(7/6)^2)-2/3

Now lets change the bit in green back to where it came from:

$\textcolor{g r e e n}{y \to {x}^{2} - \frac{7}{3} x + {\left(\frac{7}{6}\right)}^{2} \textcolor{b l u e}{- {\left(\frac{7}{6}\right)}^{2} - \frac{2}{3}}}$

Giving:

color(green)(y= (x-7/6)^2)color(blue)(-(7/6)^2-2/3#
The equals sign (=) is now back as I have included the correction.

$\textcolor{w h i t e}{\times \times \times \times} \text{----------------------------------------------}$
$\textcolor{red}{\underline{\text{Finalising the calculation}}}$

Now we can write:

$y = {\left(x - \frac{7}{6}\right)}^{2} - \left(\frac{49}{36}\right) - \frac{2}{3}$

$2 \frac{1}{36}$

$\textcolor{g r e e n}{y = {\left(x - \frac{7}{6}\right)}^{2} - \frac{73}{36}}$