Free Quadratic Equation Solver
Solve ax² + bx + c = 0 with steps.
What is the Quadratic Equation Solver?
The quadratic formula is a rite of passage in algebra, and also a genuine everyday tool for anything with a curved relationship — projectile paths, areas, and optimization problems. But plugging numbers into the formula by hand is where sign errors creep in.
This solver takes the coefficients a, b and c and finds the roots of ax² + bx + c = 0, including real and complex solutions, with the discriminant shown.
Last updated: Aug 12, 2026
What does this tool do?
You enter the three coefficients and the tool applies the quadratic formula, first computing the discriminant to decide what kind of roots the equation has.
It returns two real roots, one repeated root, or a pair of complex conjugate roots, depending on whether the discriminant is positive, zero or negative.
Key features
Real and complex roots
Handles all three discriminant cases.
Discriminant shown
See b² − 4ac and what it means.
Instant
Roots update as you change coefficients.
No sign errors
The formula is applied for you.
Private
Calculations run in your browser.
Example
Common use cases
- Students — Solve and check algebra homework.
- Teachers — Verify quadratic answers.
- Engineers — Solve equations in design work.
- Physicists — Find times or positions in motion problems.
- Developers — Solve quadratics in geometry or graphics code.
- Economists — Find break-even or optimum points.
- Test takers — Check practice problems.
- Anyone with a quadratic — Get exact roots fast.
Benefits
- It eliminates the sign slips of hand calculation.
- It handles complex roots, not just real ones.
- The discriminant explains the result.
- Nothing you enter leaves your device.
Tips
- If a is zero it isn't quadratic — it's linear, so use a different approach.
- A positive discriminant means two distinct real roots.
- A zero discriminant means one repeated real root.
- A negative discriminant means two complex conjugate roots.
- The sum of the roots is −b/a and the product is c/a — a quick sanity check.
- Enter fractions as decimals if needed.
- Double-check the sign of b, the most common error.
- The vertex x-coordinate is −b/(2a).
Common mistakes to avoid
Setting a to zero
Fix: With a = 0 the equation is linear, not quadratic — the solver needs a non-zero a.
Mis-signing b
Fix: Enter b with its sign exactly as it appears; a wrong sign flips the roots.
Expecting only real roots
Fix: A negative discriminant gives complex roots, which the tool returns.
Rounding coefficients
Fix: Enter precise values for accurate roots.
How it works
- 1
Enter coefficients
Provide a, b and c.
- 2
Get the roots
See the solutions and discriminant.
Frequently asked questions
Yes. When the discriminant is negative, it returns the complex conjugate roots.
The value b² − 4ac, which tells you whether the roots are real and distinct, repeated, or complex.
Yes. When the discriminant is negative, it returns the complex conjugate roots.
The value b² − 4ac, which tells you whether the roots are real and distinct, repeated, or complex.
Then the equation isn't quadratic — it's linear. The solver requires a non-zero a.
The roots should sum to −b/a and multiply to c/a.
Yes. Enter fractional coefficients as decimals.
No. Everything is calculated in your browser.
Conclusion
The quadratic solver gives exact roots without the sign-error risk of doing it by hand. Pair it with the circle and GCD/LCM calculators for more math work.