# Completed Square Form

The completed square form can be used to:

1. Locate the vertex of a parabola, both and coordinates;
3. 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 ?

1. 2. 3. # 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.