Completing the Square – solve or sketch a quadratic

What is Completing the Square?

Completing the Square is when either:

• we write in the form
• or we write in the form

For the simpler case where the coefficient of is 1:

1. Firstly, set as half of .
2. Secondly, expand .
3. Finally, choose so as to adjust the constant so that the original quadratic expression is obtained.

See Example 1. In contrast, if the coefficient of is not 1, first of all, remove a factor of from the original quadratic. Then perform the above on the inside of the brackets before expanding again in the final step. See Example 2.

Why is it called Completing the Square?

Sketching the graph of a quadratic can be easy if you think about the transformations that have been applied to the graph of :

• Firstly, consider the graph of . See the Board Example.
• Secondly, sketch the graph of say by shifting the graph of to the left by 1. See -transformations on the Transformations page.
• Thirdly, sketch the graph of by stretching the graph of in the -direction by a factor of 4. See -transformations on the Transformations page.
• Finally, sketching the graph of is done by shifting the graph of down by 1. See -transformations again on the Transformations page.

Examples

Write in form

Solution:

First of all, halve the coefficient of in the original quadratic (this is 4). See what happens when you set this as 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 . Hence, we must add 5 to this to get , i.e. choose to be 5. We now have:

.

Write in the form

Solution:

Students often get confused with this more complicated example. It can be made simpler by first taking out a factor of 2 and then completing the square of whatâ€™s inside the brackets:

It follows that, when expanding the final expression, we obtain the result as required:

Hence, we can see from this that and .

Videos

Completing the square with unknowns in the coefficients.