What is ‘Completing the Square’?
Completing the square is when either of the following is performed:
is written in the form
is written in the form
For the simpler case where the coefficient of is 1:
- Firstly, take
to be half of
.
- Secondly, expand
.
- Finally, choose
so as to adjust the constant so that the original quadratic expression is obtained.
See Example 1.
In contrast, the coefficient of may not be 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.
Example 1
Write in form
.
First of all, 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.
It follows that the result is but we want
and so we must add 5 to this to get
, i.e. choose
to be 5. We now have:
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 then completing the square of what’s inside the brackets:
It follows that the final term can be expanded to obtain the result as required:
Hence, we can see from this that p=q=2 and r=-13.
Sketching Quadratics
The graph of a quadratic can easily be sketched if you think about the transformations that have been applied to the graph of .
- Firstly, consider the graph of
.
- Secondly, sketch the graph of say
by shifting the graph of
to the left by 1. See x-transformations on the Transformations page.
- Thirdly, sketch the graph of
by stretching the graph of
about the x-axis by a factor of 3. See y-transformations on the Transformations page.
- Finally, the graph of
can then be sketched by shifting the graph of
down by 4. See y-transformations again on the Transformations page.
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