Quadratic Equation Solver
Quadratic equation solver: enter a, b and c to get both roots (real or complex), the discriminant and the vertex of ax² + bx + c = 0.
x = −b ± √(b² − 4ac)2a
How to use this calculator
- Type your input in the box.
- The result updates as you type.
- Use “Copy result” or “Copy link” to share it.
A quadratic equation ax² + bx + c = 0 has the solutions x = (−b ± √(b² − 4ac))/(2a). The discriminant Δ = b² − 4ac tells you how many real roots there are.
Δ > 0 gives two real roots, Δ = 0 one double root and Δ < 0 two complex conjugate roots. The solver uses a numerically stable form, so it stays accurate even when b is much larger than a and c.
Worked example
x² − 3x + 2 = 0: Δ = 9 − 8 = 1, so x = (3 ± 1)/2, giving x₁ = 2 and x₂ = 1. The vertex of the parabola is at (1.5, −0.25).
Frequently asked questions
What if a = 0?
The equation becomes linear, bx + c = 0, with the single solution x = −c/b.
How do I read complex roots?
x = p ± qi, where i = √−1. The parabola does not cross the x-axis.