Write -2x^2 + 8x - 5 in vertex form

Step by step

The question

Question: write y = −2x² + 8x − 5 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 −5 stays outside.

So y is −2 times (x² − 4x), minus 5.

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.

−2 times minus 4 is 8: the −2 multiplies both parts in the big bracket.

So y is −2 times the square, plus 8, minus 5.

4 Tidy up

8 minus 5 is 3.

So y = -2(x − 2)² + 3.

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 = 3.

So the vertex is (2, 3).

Careful!

x minus 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 −2x² + 8x − 5 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.