Completing the Square
Reference for completing the square: converting ax² + bx + c to a(x + h)² + k form to solve quadratics and find vertex form.
Includes worked examples.
Need to calculate, not just reference? Use the interactive version.
Open Completing the Square Calculator →
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
Key Notes
- Technique: ax² + bx + c is transformed into a(x + b/2a)² + (c − b²/4a). The key step is adding and subtracting (b/2a)² inside the expression to create a perfect square trinomial, then factoring it.
- Derives the quadratic formula: Applying completing the square to the general form ax² + bx + c = 0 yields x = (−b ± √(b²−4ac)) / 2a. The quadratic formula is not a separate rule — it is the result of completing the square once and keeping it in general form.
- Reveals the vertex of a parabola: The vertex form y = a(x − h)² + k shows the vertex at (h, k) directly. From ax² + bx + c, the vertex x-coordinate is h = −b/2a (the h value from completing the square).
- Discriminant meaning: b² − 4ac: Greater than zero → two distinct real roots; equal to zero → one repeated root (vertex touches x-axis); less than zero → no real roots (parabola doesn't cross the x-axis). The discriminant falls out naturally from completing the square.
- Applications: Completing the square converts conic sections to standard form, simplifies integration of quadratic expressions (partial fractions), solves optimization problems (finding the minimum cost, maximum area), and appears in deriving formulas in optics and mechanics.