Skip to content
Calculation Machine

Quadratic Formula Calculator

Roots, vertex and discriminant for ax² + bx + c.

Reviewed by the Calculation Machine team · How we check our maths

Your values

ⓘ Live results

Results
Roots
x = 3, x = 2
Discriminant
1 — two real roots
Vertex
(2.5, -0.25)
Axis of symmetry
x = 2.5

Working

  1. 1Discriminant = -5² − 4(1)(6) = 1

Quick answer

The quadratic formula gives x = (−b ± √(b² − 4ac)) ÷ 2a; the discriminant b² − 4ac tells you how many real roots exist.

The formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Worked example

Worked example

Using the values the calculator loads with.

Inputs

  • a1
  • b-5
  • c6

Result

  • Rootsx = 3, x = 2
  • Discriminant1 — two real roots
  • Vertex(2.5, -0.25)
  • Axis of symmetryx = 2.5

How to use the Quadratic Formula Calculator

  1. 1Enter the coefficients a, b and c.
  2. 2Read the roots and discriminant.
  3. 3Use the vertex to sketch the parabola.

Frequently asked questions

What does the discriminant tell me?

Positive means two real roots, zero means one repeated root, negative means the parabola never crosses the x-axis.

Can I always factor instead?

Only when roots are rational. The formula works for every quadratic, factorable or not.

About this calculator