Quadratic Equation Calculator
Solve ax² + bx + c = 0 — real or complex roots, the discriminant, the vertex and the axis of symmetry — with the quadratic formula shown.
The quadratic formula gives the roots: x = (−b ± √(b² − 4ac)) ÷ 2a.
How the quadratic equation calculator works
The quadratic formula gives the roots: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac decides the shape of the answer: positive → two real roots, zero → one repeated root, negative → two complex roots. The parabola's vertex is at x = −b ÷ 2a.
Formula: x = (−b ± √(b² − 4ac)) / 2a
Worked examples
| Inputs | Roots | Note |
|---|---|---|
| x² − 3x + 2 = 0 | x = 2 or x = 1 | roots 1 and 2 |
| x² − 4x + 4 = 0 | x = 2 | repeated root 2 |
| x² + x + 1 = 0 | x = -0.5 ± 0.866025i | complex roots |
| 2x² + 5x − 3 = 0 | x = 0.5 or x = -3 | roots 0.5 and −3 |
FAQFrequently asked questions
What is the discriminant?
b² − 4ac. Its sign tells you whether the roots are two real numbers, one repeated number or a complex pair.
What does a complex root mean?
The parabola never crosses the x-axis. The roots still exist as complex numbers a ± bi.
How do I complete the square?
Rewrite ax² + bx + c as a(x + b/2a)² + (c − b²/4a); the vertex is then obvious.
What is the vertex?
The turning point of the parabola, at x = −b/2a. It is a minimum when a > 0 and a maximum when a < 0.
Can I factorize instead?
When the roots are rational, yes: x² − 3x + 2 = (x − 1)(x − 2). The formula always works, factorizing only sometimes.
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)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.