Quadratic Equation Solver

Solve ax² + bx + c = 0 for its roots, with the discriminant, the nature of the roots, and the vertex of the parabola. Calculates in your browser.

Solve ax² + bx + c = 0.

Roots

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Discriminant-
Nature-
Vertex-

Show the math

The quadratic formula gives the roots; the discriminant says how many are real:

x = (−b ± √(b² − 4ac)) ÷ 2a

A positive discriminant means two real roots, zero means one, negative means a complex pair. All math runs in your browser.

What this does

A quadratic equation solver finds the roots of ax² + bx + c = 0, with the discriminant, the nature of the roots, and the vertex of the parabola.

How to use it

  1. Enter a, b, and c.
  2. Read the two roots.
  3. Check the discriminant.
  4. See the parabola vertex.

How it works

The discriminant b² − 4ac tells the root type: positive gives two real roots, zero one repeated root, negative a complex pair.

x = (−b ± √(b² − 4ac)) ÷ 2a

Understanding your result

The discriminant is the headline: positive gives two real roots, zero a single repeated root, and negative a complex pair with no real crossing. The roots are where the parabola meets the x-axis, and the vertex is its turning point, the minimum or maximum, which is what you want for optimisation problems rather than the roots themselves.

Example

x² − 5x + 6 = 0 solves to x = 2 and x = 3.

Sources & methodology

Last updated .

Frequently asked questions

What formula is used?

The quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a, which gives both roots of ax² + bx + c = 0.

What does the discriminant tell me?

b² − 4ac: positive means two real roots, zero means one repeated root, and negative means a complex-conjugate pair.

Is my data stored?

No. Everything is calculated locally in your browser.