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, .

For example:

- 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:

## Example 1

Half of is ; .

For brain muscle memory: ‘half the coefficient , subtract its square, remember original constant’.

## Example 2

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:

## Example 3

*‘half the coefficient of , subtract its square’*

## Example 4, odd coefficient of

# Practice

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:

Therefore .

After calculating and , (), we have

which agrees with the algebra above.