Completing the square

Concept

Completing the square requires relatively basic manipulation of quadratic equations. Students must first realise that

Every quadration equation of the form can be written as constant . This procedure generally takes 4 steps. Let us go through on how to proceed with completing the squares.

1. Express as

2. Then , we modify the expression inside the brackets so that it will fit into the form .

To do so , we add (which is the square of the 'a' coefficient in the x term above) within the brackets. To ensure that the equation stays balanced, we have to subtract outside of the brackets.

Thus , the equation will look something like this :

=

We write it as 2( ) x to clearly state the specific coefficients.

3. We would then be able to simplify the expression to .

Finally , we have managed to complete the squares.

Examples

Although these 3 steps to completing the square may seem complicated, let us try some examples to clarify our understanding.

a) Complete the squares for

First, we would take out the 3 and we would obtain From there, we attempt to turn the expression within the brackets into a square. In this example, we have to add 1/9 (square of 1/3) within the brackets. After multiplying with the 3 outside, we realize that we have to deduct 1/3 outside in order to balance the equation.

=

=

Thus, we have managed to complete the squares.

'Try it Yourself' Section

Try completing the squares with the below exercises. Practise is crucial if you wish to be good at completing the squares.

a)

b)

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