Step by step
The question
Question: write y = x² − 8x + 10 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 − 8x to be part of a bracket squared.
That is the shape of vertex form.
The 10 waits outside.
x² is a square: x by x.
−8x is two equal parts.
Half of −8 is −4.
One corner is missing: −4 times −4 is 16.
Minus times minus is plus.
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 10 is −6.
So y = (x − 4)² − 6.
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 4 is 0: at x = 4.
There the graph turns: its lowest point.
Then the square part is 0.
So y is just −6.
So the vertex is (4, −6).
It is (h, k): h is 4, k is −6.
There is no number in front of the bracket, so a is 1.
Minus 6 is plus −6, so k is −6.
Careful!
x minus 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 + 10 again.
It works!
6 Pick the choice
Our answer is y = (x − 4)² − 6.
That is choice B.
7 Why not the others
A is y = (x − 8)² − 54.
Wrong: used 8, not half of 8.
C is y = (x − 4)² + 10.
Wrong: forgot to take away the 16 we added.
D is y = (x + 4)² − 6.
Wrong: the sign: − 8x gives x − 4.
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.