Question

What is the vertex form of `y=(x – 12)(x + 8)`?

Answer 1

Solution method explained in more detail.

`color(green)(y=(x-2)^2-100`

Explanation:

Multiply the brackets giving:

`y=x^2-12x+8x-96" "->" "y=x^2-4x-96`.......Equation(1)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 1")`

Write as:

`y=(x^2-4x)-96+k`
where `k` is a correction that cancels out the error produced by the following process.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 2")`
Take the power from `x^2` and move it outside the brackets

`y=(x-4x)^2-96+k`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 3")`
Discard the `x` from `4x`

`y=(x-4)^2-96+k`
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`color(blue)("Step 3")`
halve the `-4` inside the brackets

`y=(x-2)^2-96+k" "larr" Now we need the "k`...Equation(2)
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 4")`

If you were to expand the bracket you would have:
`y=x^2-4xcolor(magenta)(+4)-96+k`

The `color(magenta)(+4)` is the error. If you compare this to Equation(1) you will find that the +4 is not in it

So we set `+4+k=0 => k=-4`

So Equation(2) becomes:

`color(brown)(y=(x-2)^2-96+kcolor(blue)(" "->" "y=(x-2)^2-96-4)`

`color(green)(y=(x-2)^2-100`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(magenta)("General case")`

`y= ax^2+bx+c " "->" "y=a(x+b/(2a))^2+c -[a(b/(2a))^2]`
`" "uarr`
`" "k`