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).
There are four choices.
First we solve it, the normal way.
Then we pick.
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 strips.
Half of 6 is 3.
One corner is missing: 3 times 3 is 9.
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 plus 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 plus 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!
6 Pick the choice
Our answer is y = (x + 3)² − 4.
That is choice B.
7 Why not the others
A is y = (x − 3)² − 4.
Wrong: the sign: + 6x gives x + 3.
C is y = (x + 3)² + 5.
Wrong: forgot to take away the 9 we added.
D is y = (x + 6)² − 31.
Wrong: used 6, not half of 6.
Let's look back.
First: complete the square.
Then: write the square.
Then: tidy up.
Then: read the vertex.
Then: check.
Then: pick the choice.
Last: why not the others.