Quadratic Formula Calculator

Enter the coefficients a, b and c to get the roots and the discriminant.

Results

Roots
x₁ = 2, x₂ = 3
Discriminant (Δ = b² − 4ac)
1
Nature of the roots
Two distinct real roots
Vertex
(2.5, -0.25)
Sum of the roots (−b/a)
5
Product of the roots (c/a)
6

A quadratic equation has the form ax² + bx + c = 0 and its graph is a parabola. The roots are the points where the parabola crosses the x-axis.

The discriminant (Δ) decides how many roots there are: positive means two distinct real roots, zero means one repeated root, and negative means no real roots — the roots are complex. The calculator solves all three cases.

How to calculate Quadratic equations

  1. Enter a, the coefficient of x²; it cannot be zero.
  2. Enter b and c; type a minus sign for negative values.
  3. Read the roots, the discriminant and the vertex of the parabola.

Quadratic equations formula

  • Δ = b² − 4ac
  • x₁,₂ = (−b ± √Δ) ÷ 2a
  • Vertex: (−b ÷ 2a, c − b² ÷ 4a)
  • Sum of roots = −b ÷ a; product of roots = c ÷ a

Quadratic equations example

For x² − 5x + 6 = 0, Δ = 25 − 24 = 1. The roots are (5 ± 1) ÷ 2, so x₁ = 2 and x₂ = 3.

Quadratic equations: common questions

What if the discriminant is negative?

The equation has no real roots and the parabola does not cross the x-axis. The roots are complex numbers of the form a ± bi.

Why can a not be zero?

With a = 0 the x² term disappears and the equation becomes linear: bx + c = 0.

What is the vertex for?

It is the lowest point of the parabola (when a is positive) or the highest (when a is negative) — the value you want in maximum-profit or minimum-cost problems.