CalcArcade.

Quadratic Formula Calculator

Enter the coefficients a, b and c of a quadratic equation to get its roots, the discriminant and the vertex of the parabola.

Your numbers

Result

How we calculate it

The solutions of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, b² − 4ac, is the discriminant.

If the discriminant is positive there are two real roots, if it is zero there is one repeated root, and if it is negative there are two complex roots. The vertex is at x = −b ÷ 2a.

Assumptions and limits

  • If a is zero the equation is linear, not quadratic.
  • Complex roots are shown in the form real ± imaginary i.

Frequently asked questions

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) ÷ (2a) gives the solutions of any quadratic equation ax² + bx + c = 0.

What does the discriminant tell me?

Its sign tells you the number and type of roots: positive for two real, zero for one repeated and negative for two complex.

How do I find the vertex?

The x coordinate of the vertex is −b ÷ 2a. Substitute it into the equation to get the y coordinate.

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Worked example (a fixed example using the starting numbers, it does not change as you type)

Inputs: a (coefficient of x²) = 2; b (coefficient of x) = -7; c (constant) = 3.

  • Root x₁ 3
  • Root x₂ 0.5
  • Discriminant 25
  • Vertex (1.75, -3.125)

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Last reviewed: . Formulas and assumptions are described on our methodology page. For information only; not professional advice.