How to Calculate Completing the Square
What is Completing the Square?
Completing the square is an algebraic technique to rewrite a quadratic ax² + bx + c in the form a(x − h)² + k. It reveals the vertex of a parabola and is used to derive the quadratic formula, solve quadratic equations, and integrate rational functions.
Step-by-Step Guide
- 1Start with ax² + bx + c
- 2Factor out a from the first two terms: a(x² + (b/a)x) + c
- 3Add and subtract (b/2a)²: a(x + b/2a)² + c − b²/4a
- 4Result: vertex form a(x − h)² + k where h = −b/2a, k = c − b²/4a
Worked Examples
Input
x² + 6x + 5
Result
(x+3)² − 4
Add/subtract (6/2)²=9: x²+6x+9−4
Input
2x² − 8x + 3
Result
2(x−2)² − 5
Factor 2, complete, vertex at (2, −5)
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