Quadratic Equation Solver
Find the roots, discriminant and vertex of any quadratic equation using the quadratic formula.
Solve any quadratic equation
The Quadratic Equation Solver finds the roots of an equation in the form ax² + bx + c = 0 using the quadratic formula. Enter the three coefficients and it instantly returns the roots — real or complex — along with the discriminant and the parabola's vertex. It is a complete companion for algebra homework and quick checks.
The quadratic formula
The roots are given by x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is the discriminant, and it determines what kind of roots you get. The tool computes everything for you, but seeing the formula helps you understand the result.
What the discriminant tells you
| Discriminant | Roots |
|---|---|
| Positive | Two distinct real roots |
| Zero | One repeated real root |
| Negative | Two complex (imaginary) roots |
How to use it
- Enter coefficients a, b, and c (a cannot be zero).
- Read the roots, which may be real or complex.
- Use the discriminant and vertex shown for graphing and analysis.
The vertex and the parabola
Every quadratic graphs as a parabola, and its turning point is the vertex at x = −b / 2a. The vertex tells you the minimum or maximum value of the expression: if a is positive the parabola opens upward (a minimum), and if a is negative it opens downward (a maximum). The roots are where the parabola crosses the x-axis.
Why quadratics matter
- Physics — projectile motion follows a quadratic path.
- Optimization — finding maximum area or minimum cost.
- Engineering and finance — modeling curves and break-even points.
- Algebra — a cornerstone topic in every math curriculum.
Private and free
All solving happens in your browser, with nothing uploaded and no limits. Solve as many equations as you like, free.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / 2a. It gives the roots of any equation in the form ax² + bx + c = 0.
What does the discriminant tell me?
If b² − 4ac is positive there are two real roots, if zero there is one, and if negative the roots are complex.
Can it solve equations with complex roots?
Yes. When the discriminant is negative, the tool returns the complex roots in a ± bi form.
Why can't a be zero?
If a is zero the equation is linear, not quadratic, so the quadratic formula does not apply.