Write 2x^2 + 12x + 7 in vertex form

Step by step

The question

Question: write y = 2x² + 12x + 7 in vertex form.

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

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

1 Take 2 out of the x parts

The x² must have just 1 in front.

So we take 2 out of the two x parts.

The 7 stays outside.

So y is 2 times (x² + 6x), plus 7.

2 Complete the square

We add 9 to fill the corner.

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

3 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.

2 times minus 9 is −18: the 2 multiplies both parts in the big bracket.

So y is 2 times the square, minus 18, plus 7.

4 Tidy up

−18 plus 7 is −11.

So y = 2(x + 3)² − 11.

This is vertex form.

5 Read the vertex

In vertex form we need x minus h.

x + 3 is x minus (−3).

So h = −3, and k = −11.

So the vertex is (−3, −11).

Careful!

x plus 3 means h is −3, not 3.

The sign in the bracket is the opposite of h.

6 Check

Let's check: open the brackets again.

(x + 3) squared is x² + 6x + 9.

We get 2x² + 12x + 7 again.

It works!

Let's look back.

First: take 2 out of the x parts.

Then: complete the square.

Then: write the square.

Then: tidy up.

Then: read the vertex.

Last: check.