Completing the Square
Transform quadratic expressions into perfect square form.
Essential for solving equations and graphing parabolas.
The Formula
ax² + bx + c = a(x + b/2a)² - b²/4a + c
Completing the square rewrites a quadratic expression as a perfect square plus a constant. This reveals the vertex of the parabola and makes solving quadratic equations straightforward.
Variables
| Symbol | Meaning |
|---|---|
| a | Coefficient of x² (must not be zero) |
| b | Coefficient of x |
| c | Constant term |
| x | The variable |
Step-by-Step Process
- Start with ax² + bx + c
- Factor out a from the first two terms: a(x² + (b/a)x) + c
- Take half of (b/a), square it: (b/2a)²
- Add and subtract that value inside the parentheses
- Result: a(x + b/2a)² + c - b²/4a
Example 1
Complete the square for x² + 6x + 2
Half of 6 is 3. Squared: 3² = 9
x² + 6x + 9 - 9 + 2
= (x + 3)² - 7
Example 2
Complete the square for 2x² - 8x + 5
Factor out 2: 2(x² - 4x) + 5
Half of -4 is -2. Squared: (-2)² = 4
2(x² - 4x + 4 - 4) + 5 = 2(x - 2)² - 8 + 5
= 2(x - 2)² - 3
When to Use It
Use completing the square when:
- Converting a quadratic to vertex form y = a(x - h)² + k
- Finding the vertex (minimum or maximum) of a parabola
- Deriving the quadratic formula itself
- Solving quadratic equations that do not factor easily