Quadratic Equation Solver
Solve ax² + bx + c = 0 with real or complex roots, an exact discriminant, the vertex and a downloadable result.
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Solve a quadratic or a linear fallback, with exact roots, decimal approximations and the quadratic’s discriminant and vertex.
How to use it
- 1Put the equation in the form ax² + bx + c = 0.
- 2Enter a, b and c separately.
- 3Solve to see the root type, values and vertex.
- 4Download JSON to keep the calculation.
Example
- Input
- a = 0.1, b = −0.3, c = 0.2
- Output
- Roots 1 and 2 with discriminant 1/100.
What happens to your data
The calculation and JSON download run in this browser tab. Anonymous workflow records exclude your coefficients and results.
Last updated November 2026
Enter the coefficients a, b and c of ax² + bx + c = 0. The solver shows the kind of solution, the discriminant and the vertex. Each root has an exact fraction or square-root expression and a decimal approximation. If a is zero, the same inputs also handle linear and constant equations.
How it works
The discriminant b² − 4ac determines whether a quadratic has two distinct real roots, one repeated real root or two complex conjugate roots. The vertex is at x = −b/(2a) and y = −(b² − 4ac)/(4a). Coefficients, the discriminant and the vertex are calculated with exact fractions. The root expressions may use an equivalent integer equation to keep the exact result readable. A square-root expression can retain square factors.
For distinct real roots that are irrational, the decimal calculation chooses the sign that avoids subtracting two close numbers. It then uses the fact that the roots multiply to c/a to recover the other root. Decimal values still have finite precision. Use the exact result to distinguish roots that round to the same displayed value.
Inputs are signed decimals with up to 12 digits before and 12 after the decimal point. Enter 0 explicitly for a missing term. The JSON includes exact root expressions, approximate real and imaginary parts, multiplicity and vertex data where applicable.
Worked example: Solve a quadratic with decimal coefficients
- Input
- a = 0.1, b = −0.3, c = 0.2
- Result
- Roots x = 1 and x = 2, discriminant 1/100, vertex (3/2, −1/40).
Common use cases
- Check a quadratic formula exercise with decimal coefficients.
- Find the vertex and determine whether a parabola opens upward or downward.
- Read complex roots with separate real and imaginary components.
- Download exact and approximate roots for a worksheet or calculation record.
Frequently asked questions
What if a is zero?＋
The equation becomes linear when b is nonzero. If a and b are both zero, every value solves the equation when c is zero and no value solves it otherwise.
What if the discriminant is negative?＋
The two roots are complex conjugates. They have the same real part and imaginary parts of opposite signs.
Why is the result sometimes written with a square root?＋
The square-root expression preserves the exact root. The decimal is a rounded representation, and the radical can retain square factors.
Can I paste an entire equation?＋
Enter a, b and c separately after putting the equation in the form ax² + bx + c = 0.
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