Completing the Square
What is completing the square?
- Completing the square is another way of writing a quadratic function
- It means rewriting in the form
- The key point is that now only occurs once in the equation
- It can be used to solve quadratic equations, sketch their graphs and to find the coordinates of the turning point
How do I complete the square?
The method used will depend on the value of the coefficient of the term in
- When
- is half of
- is
- When a ≠ 1
- First take a factor of out of the and terms
- Then continue as above
Examiner Tip
- Sometimes a question will explicitly use the phrase complete the square
- Sometimes a question will use the form without using the phrase completing the square
Worked example
Write in the form , where and are constants to be found, and is an unknown constant.
The form required is 'completing the square' (do not be put off by the , it is just a constant!).
STEP 1 - Take a factor of 3 out of the and terms - leaving avoids awkward fractions.
STEPS 2 and 3 - Complete the square on the part only. "" and "".
STEP 4 - Expand and simplify.
i.e.