From the notebook
OnTrick lesson
Completing the Square: Why the Half-Coefficient Rule Works
Rewrite a quadratic in completed-square form and understand the half-the-coefficient rule instead of memorising it blindly.
What you are trying to create
Completing the square rewrites x² + bx + c as (x + p)² + q. This can make turning points, minimum values and equation solving easier to see.
Where the “halve it” rule actually comes from
Expand (x+p)² and you get x² + 2px + p². To match an x-coefficient of b, you need 2p=b, so p=b/2.
Worked example
Rewrite:
x² + 6x + 5.- Half 6 to get 3.
- Start with
(x+3)². - Expanding that gives
x²+6x+9, which is 4 too large. - Subtract 4.
x² + 6x + 5 = (x + 3)² − 4
Negative middle term
For x² − 8x + 1, half of −8 is −4:
x² − 8x + 1 = (x − 4)² − 15
If x² has a coefficient
Factor it from the x² and x terms first. For 2x²+8x+1:
2(x² + 4x) + 1
= 2[(x+2)² − 4] + 1
= 2(x+2)² − 7
Expand it once to catch a sign error
Expanding your final form should return the original quadratic exactly. This is the fastest self-check.
Try one yourself
In x² + 10x + 7 = (x + 5)² + q, what is q?
Keep going
For another problem using related ideas, return to the Algebra & Equations page.