Write 3x^2 + 6x - 1 in vertex form

Step by step

The question

Question: write y = 3x² + 6x − 1 in vertex form.

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

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

1 Take 3 out of the x parts

The x² must have just 1 in front.

So we take 3 out of the two x parts.

The −1 stays outside.

So y is 3 times (x² + 2x), minus 1.

2 Complete the square

We add 1 to fill the corner.

And we take 1 away straight after, so nothing changes.

3 Write the square

The full square is x + 1 on each side.

So these three parts are (x + 1) squared.

We write the bracket squared.

The rest stays.

3 times minus 1 is −3: the 3 multiplies both parts in the big bracket.

So y is 3 times the square, minus 3, minus 1.

4 Tidy up

−3 minus 1 is −4.

So y = 3(x + 1)² − 4.

This is vertex form.

5 Read the vertex

In vertex form we need x minus h.

x + 1 is x minus (−1).

So h = −1, and k = −4.

So the vertex is (−1, −4).

Careful!

x plus 1 means h is −1, not 1.

The sign in the bracket is the opposite of h.

6 Check

Let's check: open the brackets again.

(x + 1) squared is x² + 2x + 1.

We get 3x² + 6x − 1 again.

It works!

Let's look back.

First: take 3 out of the x parts.

Then: complete the square.

Then: write the square.

Then: tidy up.

Then: read the vertex.

Last: check.