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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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