Quadratic equation solver
Solve quadratic equations (ax² + bx + c = 0) and find real or complex roots, vertex, axis of symmetry and discriminant.
- Free, no sign-up
- No data stored
- Reviewed 2026-01-15
Your details
Results
Two real roots
x₁ = 5 · x₂ = -2
1x² − 3x − 10 = 0
- x₁
- 5
- x₂
- -2
- Factored form
- a(x − x₁)(x − x₂)
- Discriminant
- 49
- Vertex
- (1.5, -12.25)
- Axis of symmetry
- x = 1.5
- Opens
- Upwards
- Y intercept
- -10
Δ = b² − 4ac
Working
Δ = b² − 4ac = (-3)² − 4·1·-10 = 49
x = (−b ± √Δ) / 2aAbout the Quadratic Equation Solver
Enter coefficients a, b and c to solve ax² + bx + c = 0. The calculator shows the discriminant, both roots (real or complex) and the vertex coordinates. It also identifies whether the parabola opens up or down.
How to use the Quadratic Equation Solver
- Enter coefficient a (must not be zero).
- Enter coefficients b and c.
- View the discriminant, roots and vertex.
Formula and method
x = (−b ± √(b²−4ac)) / 2a
Frequently asked questions
What if the discriminant is negative?
The equation has two complex (imaginary) roots. They are displayed in a + bi form.
What does the vertex represent?
The vertex is the minimum (a > 0) or maximum (a < 0) point of the parabola y = ax² + bx + c.
Can a be zero?
No. If a = 0, the equation is linear (bx + c = 0), not quadratic. Enter it as a linear equation: x = −c/b.