Quadratic Equation Solver
Solve any quadratic equation and find its roots and vertex.
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How it works
Enter the coefficients a, b, and c of ax² + bx + c = 0. The solver computes the discriminant (b² − 4ac), which tells you the nature of the roots: positive gives two real roots, zero gives one repeated root, and negative gives a pair of complex conjugates. It then applies the quadratic formula to find the roots, and also reports the vertex of the parabola. Complex results are shown in a ± bi form. All math runs in your browser; a must be non-zero.
x = (−b ± √(b² − 4ac)) / 2a
Common use cases
- Checking homework answers for quadratic equations.
- Finding where a parabola crosses the x-axis.
- Getting the vertex (turning point) of a quadratic function.
Frequently asked questions
What does the discriminant tell me?
The discriminant b² − 4ac determines the roots: greater than zero means two distinct real roots, exactly zero means one repeated real root, and less than zero means two complex conjugate roots.
Can it return complex roots?
Yes. When the discriminant is negative, the solver returns the complex roots in the form p ± qi rather than just saying there is no real solution.
Why must a not be zero?
If a is zero the equation is linear, not quadratic, and the x² term disappears. The solver asks for a non-zero a and points you to basic algebra for the linear case.