Step by step
The question
Question: write y = x² + 8x + 3 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 + 8x to be part of a bracket squared.
That is the shape of vertex form.
The 3 waits outside.
x² is a square: x by x.
8x is two equal strips.
Half of 8 is 4.
One corner is missing: 4 times 4 is 16.
We add 16 to fill the corner of the picture.
Then the picture is a full square.
And we take 16 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 4: x + 4.
So these three parts are (x + 4) squared.
Opened, it gives the same: x squared, two strips, and the corner.
We write the bracket squared.
The rest stays.
3 Tidy up
−16 plus 3 is −13.
So y = (x + 4)² − 13.
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 plus 4 is 0: at x = −4.
There the graph turns: its lowest point.
Then the square part is 0.
So y is just −13.
So the vertex is (−4, −13).
It is (h, k): h is −4, k is −13.
There is no number in front of the bracket, so a is 1.
Minus 13 is plus −13, so k is −13.
Careful!
x plus 4 is 0 when x is −4, not 4.
The sign in the bracket is the opposite.
5 Check
Let's check: open the brackets again.
(x + 4) squared is x² + 8x + 16.
We get x² + 8x + 3 again.
It works!
Let's look back.
First: complete the square.
Then: write the square.
Then: tidy up.
Then: read the vertex.
Last: check.