Completing the Square Calculator
Rewrite ax² + bx + c as a(x − h)² + k — the vertex form, which shows the turning point directly.
Completing the square rewrites a quadratic so the variable appears once instead of twice.
How the completing the square calculator works
Completing the square rewrites a quadratic so the variable appears once instead of twice. Factor a out of the first two terms, halve the resulting coefficient of x, square it, and add and subtract that amount.
The payoff is that (h, k) is the vertex of the parabola — the minimum when a is positive, the maximum when it is negative. Reading it off the vertex form takes no work, where finding it from the standard form takes calculus or the formula −b/2a.
This is also where the quadratic formula comes from: complete the square on the general quadratic and solve, and the formula falls out.
Formula: ax² + bx + c = a(x + b/2a)² + (c − b²/4a)
Worked examples
| Inputs | Vertex form | Note |
|---|---|---|
| 2x² − 12x + 23 | 2(x - 3)² + 5 | 2(x − 3)² + 5 |
| x² + 6x + 5 | (x + 3)² - 4 | (x + 3)² − 4 |
| A downward parabola | -(x - 2)² + 3 | maximum of 3 at x = 2 |
FAQFrequently asked questions
What is vertex form?
a(x − h)² + k, where (h, k) is the turning point of the parabola.
How do I complete the square?
Halve the coefficient of x, square it, then add and subtract that amount — adjusting for the leading coefficient first.
Why bother?
Because the vertex, the axis of symmetry and the range all read straight off the vertex form, and none of them are visible in the standard form.
What is h?
−b/2a — the x-coordinate of the vertex and of the axis of symmetry.
How does this give the quadratic formula?
Complete the square on the general ax² + bx + c = 0 and solve for x; the formula is the result.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.