Step by step
The question
Question: write y = x² + 6x + 5 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
−9 plus 5 is −4.
So y = (x + 3)² − 4.
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 = −4.
So the vertex is (−3, −4).
Careful!
x plus 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 + 5 again.
It works!
Let's look back.
First: complete the square.
Then: write the square.
Then: tidy up.
Then: read the vertex.
Last: check.