Completing the square rewrites a quadratic so that x appears only once. Halve the coefficient of x to get p, then (x + p)² gives x² + bx + p². Subtract the extra p² and add c.
For x² − 6x − 12: half of −6 is −3, so (x − 3)² = x² − 6x + 9. We need −12, not +9, so subtract 21: x² − 6x − 12 = (x − 3)² − 21. The turning point of the graph is (3, −21), and the equation can be solved by square-rooting.
Worked example
Write x² + 10x + 7 in the form (x + p)² + q.
- Half of 10 is 5: (x + 5)² = x² + 10x + 25.
- 7 − 25 = −18.
Answer:(x + 5)² − 18
Key words
- completed square form
- (x + p)² + q
- turning point
- the lowest or highest point of the curve
- halve
- divide by 2
Practise this
New questions every time.
Common Core-referenced: HSA-REI.B.4a