## Completing the Square

Completing the square is when a quadratic of the form is rewritten in the form .

Let us consider the simpler case where .

**Example 1: Write in form .**

Take the coefficient of in the original quadratic (this is 4) and halve it – see what happens when you choose to be this value and expand :

Now we can see why we should halve the number as you end up with two lots of it in the expansion.

The result is but we want and so we must add 5 to this to get . It follows that:

We can also think of as and adding 9 to both obtains the required result.

We now consider an example of completing the square where .

**Example 2 – Write in the form .**

Students often get confused with this more complicated example. It can be made simpler to first taking out a factor of 2 and completing the square of what’s inside the brackets:

The final term can be expanded as follows to obtain the result as required: