Write x^2 - 6x in vertex form

Step by step

The question

Question: write y = x² − 6x in vertex form.

Vertex form is a times (x minus h) squared, plus k.

The vertex, the turning point, is (h, k).

1 Complete the square

We add 9 to fill the corner.

And we take 9 away straight after, so nothing changes.

2 Write the square

The full square is x − 3 on each side.

So these three parts are (x − 3) squared.

We write the bracket squared.

The rest stays.

3 Tidy up

So y = (x − 3)² − 9.

This is vertex form.

4 Read the vertex

In vertex form we need x minus h.

x − 3 is x minus 3.

So h = 3, and k = −9.

So the vertex is (3, −9).

Careful!

x minus 3 means h is 3, not −3.

The sign in the bracket is the opposite of h.

5 Check

Let's check: open the brackets again.

(x − 3) squared is x² − 6x + 9.

We get x² − 6x again.

It works!

Let's look back.

First: complete the square.

Then: write the square.

Then: tidy up.

Then: read the vertex.

Last: check.