Quadratic Equation Solver
Solve ax² + bx + c = 0 — real or complex roots, discriminant, and vertex
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Quick summary
Enter the coefficients a, b, and c to solve any quadratic equation with the quadratic formula, x = (−b ± √(b²−4ac)) / 2a. You get the roots (real or complex), the discriminant, and the parabola's vertex and axis of symmetry. For x² − 5x + 6 = 0 the roots are 2 and 3, the discriminant is 1, and the vertex is (2.5, −0.25). The discriminant's sign tells you the nature: positive → two real roots, zero → one, negative → a complex pair.
Reviewed by Simple Calculators TeamLast reviewed
How to use this calculator
- Enter the inputs requested by the calculator.
- Choose the units or options that match your situation and review the assumptions.
- Run or update the calculation to see the result.
- Review the result, breakdown, and formulas shown on the page.
- Check the inputs, units, dates, and limitations before saving or sharing the result.
Frequently asked questions
How do I solve a quadratic equation?
Use the quadratic formula: x = (−b ± √(b²−4ac)) / 2a, where a, b, and c are the coefficients of ax² + bx + c = 0. Compute the discriminant b²−4ac first — its sign tells you whether the roots are real or complex — then apply the formula. This solver does all of it and also shows the vertex.
What is the discriminant?
The discriminant is b²−4ac, the part under the square root in the quadratic formula. If it's positive there are two distinct real roots; if zero, one repeated real root; if negative, two complex-conjugate roots. It tells you the nature of the solutions before you finish solving.
What are complex roots?
When the discriminant is negative, the square root of a negative number produces an imaginary component, so the roots take the form a ± bi (where i = √−1). The parabola never crosses the x-axis in that case. This solver reports the real and imaginary parts.
What is the vertex of a parabola?
The vertex is the turning point of the parabola y = ax² + bx + c, located at x = −b/2a. It's the minimum if a > 0 (opens upward) or the maximum if a < 0 (opens downward). The vertical line through it, x = −b/2a, is the axis of symmetry.
Key terms
- Quadratic Formula
- x = (−b ± √(b²−4ac)) / 2a — solves any quadratic.
- Discriminant
- b²−4ac; its sign determines whether roots are real or complex.
- Vertex
- The turning point of the parabola, at x = −b/2a.
- Root
- A solution where the equation equals zero (an x-intercept if real).
Methodology, sources & limitations
This page shows the formula, assumptions, and limitations used for this estimate. Rates, thresholds, and other time-sensitive inputs can change, so verify them against the relevant official source before relying on a result.
For our review approach and the primary sources we consult, see the methodology and source policy. If you spot an outdated figure or formula, please contact us with a correction.