Step by step
The question
Question: write y = x² − 10x + 30 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 25 to fill the corner.
And we take 25 away straight after, so nothing changes.
2 Write the square
The full square is x − 5 on each side.
So these three parts are (x − 5) squared.
We write the bracket squared.
The rest stays.
3 Tidy up
−25 plus 30 is 5.
So y = (x − 5)² + 5.
This is vertex form.
4 Read the vertex
In vertex form we need x minus h.
x − 5 is x minus 5.
So h = 5, and k = 5.
So the vertex is (5, 5).
Careful!
x minus 5 means h is 5, not −5.
The sign in the bracket is the opposite of h.
5 Check
Let's check: open the brackets again.
(x − 5) squared is x² − 10x + 25.
We get x² − 10x + 30 again.
It works!
Let's look back.
First: complete the square.
Then: write the square.
Then: tidy up.
Then: read the vertex.
Last: check.