Question

What is the vertex form of `y=(x – 12)(x + 4)`?

Answer 1

`y=(x-4)^2-64`

Explanation:

First, distribute the binomials terms.

`y=x^2+4x-12x-48`

`y=x^2-8x-48`

From here, complete the square with the first two terms of the quadratic equation.

Recall that vertex form is `y=a(x-h)^2+k` where the vertex of the parabola is at the point `(h,k)`.

`y=(x^2-8xcolor(red)(+16))-48color(red)(-16)`

Two things just happened:

The `16` was added inside the parentheses so that a perfect square term will be formed. This is because `(x^2-8x+16)=(x-4)^2`.

The `-16` was added outside the parentheses to keep the equation balanced. There is a net change of `0` now thanks to the addition of `16` and `-16`, but the face of the equation is changed.

Simplify:

`y=(x-4)^2-64`

This tells us that the parabola has a vertex at `(4,-64)`. graph{(x-12)(x+4) [-133.4, 133.5, -80, 40]}