Quadratic Equation Solver
Solve ax² + bx + c = 0 — real roots, complex roots, vertex and more.
All calculations happen locally in your browser — nothing is uploaded, stored or tracked.
Enter a, b and c
The coefficients of ax² + bx + c = 0.
Calculate
The solver evaluates the discriminant and both roots.
Read the summary
Nature of the roots, vertex and axis of symmetry.
What Is a Quadratic Equation Solver?
A quadratic equation is any equation of the form ax² + bx + c = 0. This solver applies the quadratic formula to find both roots, and goes further: it reports the discriminant, which tells you the nature of the roots, plus the vertex and axis of symmetry of the parabola the equation describes.
When the discriminant is negative the equation has no real roots — the solver returns the two complex roots instead, so you always get a complete answer.
The Quadratic Formula
If Δ > 0 there are two distinct real roots; if Δ = 0 there is one repeated real root; if Δ < 0 there are two complex conjugate roots. The vertex is at x = −b/(2a), and its y-coordinate is the value of the expression there.
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Frequently Asked Questions
How do I solve a quadratic equation?
Put the equation in the form ax² + bx + c = 0, then apply x = (−b ± √(b² − 4ac)) / 2a.
What is the discriminant?
It is b² − 4ac. Positive means two real roots, zero means one repeated root, negative means two complex roots.
What if the discriminant is negative?
The equation has no real solutions, only complex ones. This solver returns them as a ± bi.
How do I find the vertex of a parabola?
The x-coordinate is −b/(2a); substitute it back into the equation for the y-coordinate.
What does a = 0 mean?
The equation is not quadratic — it becomes linear (bx + c = 0), solved as x = −c/b.