The completed square form can be used to:
- Locate the vertex of a parabola, both and coordinates;
- Solve a quadratic equation;
- Calculate the inverse of a quadratic function – discussed in the grade 12 curriculum.
A Perfect Square
is known as a perfect square because it can be expressed as a single term squared:
We can confirm this by multiplying out the brackets on the right hand side:
is almost a perfect square. It is just one unit more than . It can be written:
‘Completing the square’ is to assume that your quadratic expression is a perfect square, and then to make an adjustment so that the constant term is correct.
Remember that in general, a perfect square has the form . (Assuming .)
Suppose the following are perfect squares – what is the constant ?
Completing the Square when
When the coefficient of the term we can write the quadratic expression in completed square form using this process:
- half the coefficient , and write ;
- subtract the square of , to write ;
- Add the original constant and simplify, .
- half the coefficient , write . This is equal to ;
- subtract the square of : write This is equal to ;
- Add the original constant and simpilfy: This is equal to .
These three steps can all be done in one line:
Half of is ; .
For brain muscle memory: ‘half the coefficient , subtract its square, remember original constant’.
Half of , is , the square of is .
To check that the right hand side is equal to the left hand side, lets multiply out and simplify:
‘half the coefficient of , subtract its square’
Example 4, odd coefficient of
Use this applet to develop muscle memory for the complete the square process:
When the coefficient of is not 1
How we complete the square depends if we are solving an equation or rewriting an expression.
If we have an equation, we can divide both sides by the coefficient :
(remember that ). The rest of this work is detailed in Example 4 on another page.
If we have an expression, we need to factor the coefficient of :
We can then complete the square within the large brackets:
This algebra is fairly tricksome and can take some practice to master. However, in a pinch, one might wish to use the following:
Using the Vertex to Complete the Square
On another page, we argue that if , the vertex has coordinates . We can find the values and using any ‘find the vertex’ method.
On a different page, we see that
Considering our example above, , . Therefore, .
To find we simply need to substitute to the expression:
After calculating and , (), we have
which agrees with the algebra above.