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 want x² − 6x to be part of a bracket squared.
That is the shape of vertex form.
The 5 waits outside.
x² is a square: x by x.
−6x is two equal parts.
Half of −6 is −3.
One corner is missing: −3 times −3 is 9.
Minus times minus is plus.
We add 9 to fill the corner of the picture.
Then the picture is a full square.
And we take 9 away straight after, so nothing changes.
We put the first three parts together: they make the full square.
2 Write the square
In the picture, each side of the full square is x and −3: x − 3.
So these three parts are (x − 3) squared.
Opened, it gives the same: x squared, two strips, and the corner.
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
A square is never below 0: a number times itself is never minus.
So it is smallest, 0, when x minus 3 is 0: at x = 3.
There the graph turns: its lowest point.
Then the square part is 0.
So y is just −4.
So the vertex is (3, −4).
It is (h, k): h is 3, k is −4.
There is no number in front of the bracket, so a is 1.
Minus 4 is plus −4, so k is −4.
Careful!
x minus 3 is 0 when x is 3, not −3.
The sign in the bracket is the opposite.
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.