What does the discriminant tell me?

The discriminant (b² − 4ac) determines the type of roots. Positive means two distinct real roots, zero means one repeated real root (the parabola touches the x-axis once), and negative means two complex-conjugate roots (the parabola never crosses the x-axis).

What happens if a = 0?

With a = 0 the equation isn't quadratic anymore — it becomes the linear equation bx + c = 0, which the solver detects automatically and solves for x = −c/b instead.

What is the vertex, and why does it matter?

The vertex is the parabola's turning point, at x = −b/2a. It's the minimum point if a is positive or the maximum if a is negative, and it's exactly halfway between two real roots — useful for sketching the graph or for optimization problems.

Is my equation sent anywhere?

No. All calculations run locally in your browser with JavaScript — nothing is transmitted or stored.

QUADRATIC EQUATION SOLVER

Solve ax² + bx + c = 0 instantly, including complex roots. Nothing leaves your browser.

a·x² + b·x + c = 0

x₁2

x₂1

Discriminant (b²−4ac): 1Vertex: (1.5, -0.25)

How to use the quadratic equation solver

1
Enter the coefficients a, b, and c from your equation ax² + bx + c = 0.
2
The solver instantly applies the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.
3
Read the roots — two real roots, one repeated root, or a complex-conjugate pair, depending on the discriminant.
4
Check the discriminant and vertex coordinates shown below the result for extra context.

Questions

What does the discriminant tell me?: The discriminant (b² − 4ac) determines the type of roots. Positive means two distinct real roots, zero means one repeated real root (the parabola touches the x-axis once), and negative means two complex-conjugate roots (the parabola never crosses the x-axis).
What happens if a = 0?: With a = 0 the equation isn't quadratic anymore — it becomes the linear equation bx + c = 0, which the solver detects automatically and solves for x = −c/b instead.
What is the vertex, and why does it matter?: The vertex is the parabola's turning point, at x = −b/2a. It's the minimum point if a is positive or the maximum if a is negative, and it's exactly halfway between two real roots — useful for sketching the graph or for optimization problems.
Is my equation sent anywhere?: No. All calculations run locally in your browser with JavaScript — nothing is transmitted or stored.