Completing the square
Objective. Rewrite a quadratic as (x + p)² + q and find the turning point.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Halve the coefficient of x to get p, then subtract p² to keep the expression equal. The turning point is (−p, q).
Worked example
Write x² + 6x + 1 in completed square form.
- 1Half of 6 is 3, so start with (x + 3)².
- 2(x + 3)² = x² + 6x + 9, which is 8 too many.
- 3So x² + 6x + 1 = (x + 3)² − 8.
Answer: (x + 3)² − 8, turning point (−3, −8)
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