Completing the square, step by step
Completing the square rewrites a quadratic as a square plus a number.
The question
Complete the square: x² − 6x + 24
x² − 6x + 24
−6 ÷ 2 = −3
(x − 3)² = x² − 6x + 9
x² − 6x + 24 = (x − 3)² + 15
- Halve the middle coefficient. Half of −6 is −3, and that number goes straight into the bracket — sign and all.
- But (x − 3)² is not the expression yet. Multiply it out and it brings a 9 along with it, and the original has 24 there instead.
- So give the 9 back: 24 − 9 = 15. That is the number that goes after the bracket.
How it works.
- 01
Halve the middle coefficient. Half of −6 is −3, and that number goes straight into the bracket — sign and all.
- 02
But (x − 3)² is not the expression yet. Multiply it out and it brings a 9 along with it, and the original has 24 there instead.
- 03
So give the 9 back: 24 − 9 = 15. That is the number that goes after the bracket.