Completing the Square Calculator
Rewrite any quadratic ax² + bx + c into vertex form a(x − h)² + k with step-by-step work.
Finds the vertex, axis of symmetry, and roots.
Completing the Square
Completing the square is an algebraic technique that rewrites a quadratic expression from standard form into vertex form. This reveals the vertex of the parabola directly — without solving the quadratic formula.
Standard form vs Vertex form:
| Form | Expression |
|---|---|
| Standard Form | ax^2 + bx + c |
| Vertex Form | a(x - h)^2 + k |
The formulas:
| Property | Formula |
|---|---|
| h (x-coordinate of vertex) | h = -b / (2a) |
| k (y-coordinate of vertex) | k = c - b^2 / (4a) |
| Axis of Symmetry | x = h |
| Discriminant | D = b^2 - 4ac |
Step-by-step method for ax^2 + bx + c:
- Factor out a from the first two terms: a(x^2 + b/a * x) + c
- Add and subtract (b/2a)^2 inside the parentheses
- The first three terms form a perfect square: a(x + b/2a)^2 - b^2/(4a) + c
- Simplify to get a(x - h)^2 + k
Worked example — x^2 - 6x + 5:
- h = -(-6) / (2 × 1) = 3
- k = 5 - 36/4 = 5 - 9 = -4
- Vertex form: (x - 3)^2 - 4
- Vertex: (3, -4), Axis of symmetry: x = 3
- Roots: D = 36 - 20 = 16 > 0, so x = (6 ± 4) / 2 → x = 5 or x = 1
Why use this technique? Completing the square is the foundation for deriving the quadratic formula. It also makes it easy to find the maximum or minimum value of a quadratic (which is k when a > 0 for minimum, a < 0 for maximum), and it is essential in integration, conic sections, and converting circle equations to standard form.