Quadratic Formula — ax² + bx + c = 0
x = (−b ± √(b²−4ac)) / 2a
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.
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Tip: Completing the square is how calculators and computers find roots of quadratics internally. It's also used in physics to find the time of maximum height in projectile motion (find vertex of parabola).
- 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
x² + 6x + 5=(x+3)² − 4Add/subtract (6/2)²=9: x²+6x+9−4
2x² − 8x + 3=2(x−2)² − 5Factor 2, complete, vertex at (2, −5)
| Step | Action | Example: x²+6x+5 |
|---|---|---|
| 1 | Move constant to right | x²+6x = −5 |
| 2 | Add (b/2)² to both sides | x²+6x+9 = −5+9 = 4 |
| 3 | Factor left side | (x+3)² = 4 |
| 4 | Square root both sides | x+3 = ±2 |
| 5 | Solve for x | x = −1 or x = −5 |
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Fun Fact
The quadratic formula x = (−b ± √(b²−4ac)) / 2a is derived by completing the square on the general quadratic ax²+bx+c=0. Every time you use the formula, you're implicitly completing the square.
References
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