Question

What is the vertex form of `y=7x^2-9x-32`?

Answer 1

`y_("vertex form")=7(x-9/14)^2-977/28`

Explanation:

Given: `y=7x^2-9x-32`......................(1)

Write as:

`y=7(x^2-9/7x)-32`

Now write as

`y=7(x-[1/2xx9/7])^2-32 color(blue)(+"correction")`

`y=7(x-9/14)^2-32color(blue)(+"correction")`..........................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider `7(x-9/14)^2`
This gives: `7(x^2-9/7x+81/196)`
We need the `7(x^2-9/7x)` but the `7(+81/196)` is an extra value we need to get rid of. This is why we have a correction. In this case the correction value is:`color(blue)(7( -81/196)=-81/28)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So equation (2) becomes:

`y=7(x-9/14)^2-32color(blue)(+(-81/28))`..........................(2_a)

`y=7(x-9/14)^2-977/28`

`("So " x_("vertex")=(-1)xx(-9/14)color(white)(..)=+9/14)`