Watch out
Forgetting to subtract the square that was added.
Back to National 5 Mathematics
SQA National 5 Mathematics
Completing the square
Rewriting quadratics to reveal the turning point.
Formula reference
Check the National 5 rules and formulae linked to this topic.
Which method?
Match exam clues to a suitable method.
Before you start
- Know how to expand (x + a)²
- Halve the coefficient of x.
- Remember to balance the extra square term.
Explanation
Completing the square rewrites a quadratic to show the turning point.
For x² + bx + c, halve b, square it, then adjust the constant.
The form (x + p)² + q makes the minimum or maximum point easier to identify.
Key formulae and rules
- x² + bx + c = (x + b2)² + c − b2²
Check
Compare your answer with the size you expected from the question.
Exam tip
In (x − p)² + q, the turning point is (p, q), not (-p, q)
Calculator tip
Use brackets for fractions, powers and square roots, then round only at the final line.
Worked examples
Worked example 1
Positive x term
Write x² + 6x + 11 in completed square form
- Half of 6 is 3.
- (x + 3)² = x² + 6x + 9
- Add 2 to reach 11.
Answer: x² + 6x + 11 = (x + 3)² + 2
Worked example 2
Negative x term
Write x² − 8x + 5 in completed square form
- Half of −8 is −4.
- (x − 4)² = x² − 8x + 16
- Subtract 11 to reach 5.
So: (x − 4)² − 11
Worked example 3
Find a turning point
Find the turning point of y = x² − 10x + 21.
- Complete the square: (x − 5)² − 25 + 21
- y = (x − 5)² − 4
Answer: Turning point is (5, −4).
Watch out
- Forgetting to subtract the square that was added.
- Halving the constant instead of the x coefficient.
- Writing the turning point sign the wrong way round.