Step by step
The question
Question: write y = x² + 4x + 7 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 4 to fill the corner.
And we take 4 away straight after, so nothing changes.
2 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 Tidy up
−4 plus 7 is 3.
So y = (x + 2)² + 3.
This is vertex form.
4 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 plus 2 means h is −2, not 2.
The sign in the bracket is the opposite of h.
5 Check
Let's check: open the brackets again.
(x + 2) squared is x² + 4x + 4.
We get x² + 4x + 7 again.
It works!
Let's look back.
First: complete the square.
Then: write the square.
Then: tidy up.
Then: read the vertex.
Last: check.