Write -3x^2 - 12x - 7 in vertex form

Step by step

The question

Question: write y = −3x² − 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 −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 −7 stays outside.

So y is −3 times (x² + 4x), minus 7.

2 Complete the square

We add 4 to fill the corner.

And we take 4 away straight after, so nothing changes.

3 Write the square

The full square is x + 2 on each side.

So these three parts are (x + 2) squared.

We write the bracket squared.

The rest stays.

−3 times minus 4 is 12: the −3 multiplies both parts in the big bracket.

So y is −3 times the square, plus 12, minus 7.

4 Tidy up

12 minus 7 is 5.

So y = -3(x + 2)² + 5.

This is vertex form.

5 Read the vertex

In vertex form we need x minus h.

x + 2 is x minus (−2).

So h = −2, and k = 5.

So the vertex is (−2, 5).

Careful!

x plus 2 means h is −2, not 2.

The sign in the bracket is the opposite of h.

6 Check

Let's check: open the brackets again.

(x + 2) squared is x² + 4x + 4.

We get −3x² − 12x − 7 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.