Question

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

Answer 1

`color(green)(y= (x-7/6)^2-73/36)`
Notice I have kept it in fractional form. This is to maintain precision.

Explanation:

Divide through out by 3 giving:
`y=x^2-7/3x-2/3`

British name for this is: completing the square

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

`color(brown)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")`
`color(brown)("Consider the part that is: "x^2-7/3x)`
`color(brown)("Take the"(-7/3)"and halve it. So we have"1/2 xx(-7/3)=(-7/6))`
`color(brown)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")`
Now write: `y-> (x-7/6)^2-2/3`

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.
`color(white)(xxxxxxxx)"----------------------------------------------"`

`color(red)(underline("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.
`color(white)(xxxxxxxx)"----------------------------------------------"`

`color(red)(underline("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:

`color(green)(y->x^2- 7/3 x+(7/6)^2color(blue)(-(7/6)^2-2/3))`

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.

`color(white)(xxxxxxxx)"----------------------------------------------"`
`color(red)(underline("Finalising the calculation"))`

Now we can write:

`y= (x-7/6)^2-(49/36)-2/3`

`2 1/36`

`color(green)(y= (x-7/6)^2-73/36)`