Quadratic Equation Calculator

MathLast updated: August 17, 2026

A quadratic equation has the form ax² + bx + c = 0, and its solution depends on the discriminant (Δ = b² − 4ac): if Δ > 0 there are two distinct real roots, if Δ = 0 there is a single repeated root, and if Δ < 0 there are no real roots at all — the roots are complex numbers.

This calculator finds the discriminant and the roots, reports the vertex and the axis of symmetry of the parabola, explains the solution step by step in plain English, and plots the graph. When Δ < 0 it also gives the complex roots, which makes it ideal for checking homework and analyzing functions.

Cannot be 0 — with a = 0 the equation is no longer quadratic.

Formula

ax² + bx + c = 0 (a ≠ 0)
Δ = b² − 4ac
x₁,₂ = (−b ∓ √Δ) / (2a)
Vertex: x = −b / 2a, y = c − b² / 4a
Vieta: x₁ + x₂ = −b/a, x₁ × x₂ = c/a

Most errors come from signs: when b is negative, −b becomes positive, and b² is always positive. For b = −5, for example, Δ = (−5)² − 4ac uses (−5)² = 25, not −25.

How to Calculate

  1. Rewrite your equation in the form ax² + bx + c = 0 — move every term to the left-hand side.
  2. Enter the coefficients a, b, and c with their signs (a ≠ 0).
  3. Read the discriminant, the roots, and the vertex in the results section.
  4. Copy the step-by-step solution from the interpretation line into your notes, and confirm the roots visually on the graph.

Worked Examples

Two real roots: x² − 5x + 6 = 0

Δ = b² − 4ac = (−5)² − 4 × 1 × 6 = 1. Since Δ > 0 there are two distinct real roots: x₁ = 2 and x₂ = 3. The vertex is (2.50; -0.25) and the axis of symmetry is x = 2.50.

Discriminant (Δ): 1 · Roots: x₁ = 2, x₂ = 3 · Vertex: (2.50; -0.25)

Repeated root: x² + 4x + 4 = 0

Δ = b² − 4ac = 4² − 4 × 1 × 4 = 0; from the identity (x + 2)² = 0 the repeated root is x = -2. The parabola is tangent to the x-axis at the vertex (-2; 0).

Discriminant (Δ): 0 · Repeated (double) root: x₁ = x₂ = -2 · Vertex: (-2; 0)

No real roots: x² + x + 1 = 0

Δ = b² − 4ac = 1² − 4 × 1 × 1 = -3 is negative: the parabola never crosses the x-axis, so there is no real root. The complex roots are x = -0.500 ± 0.866i and the vertex is (-0.50; 0.75).

Discriminant (Δ): -3 · Complex roots: x = -0.500 ± 0.866i · Vertex: (-0.50; 0.75)

Frequently Asked Questions

What is the discriminant (delta) used for?
Δ = b² − 4ac tells you how many real solutions the equation has before you compute any root: two distinct roots when Δ > 0, a single repeated root when Δ = 0, and no real root when Δ < 0. Because it sits under the square root in the quadratic formula, its sign decides everything.
Why can the coefficient a not be 0?
With a = 0 the x² term disappears and the equation becomes the first-degree equation bx + c = 0; the denominator 2a in the quadratic formula also becomes undefined. Such an equation has the single solution x = −c/b, and this calculator warns you in that case.
What are the roots when Δ < 0?
There is no solution among the real numbers; the roots are a conjugate pair of complex numbers of the form x = −b/2a ± (√(−Δ)/2a)i. Graphically this means the parabola never crosses the x-axis. The calculator computes and displays these roots as well.
How do you find the vertex, and what does it mean?
The vertex has x-coordinate x = −b/2a and y-coordinate y = c − b²/4a. When a > 0 the parabola opens upward and the vertex is the minimum; when a < 0 it opens downward and the vertex is the maximum. The vertex is the answer to optimization questions such as maximizing profit.
What are Vieta's formulas?
They give the sum and the product of the roots without finding the roots one by one: x₁ + x₂ = −b/a and x₁ × x₂ = c/a. They are ideal for a quick check: in x² − 5x + 6 = 0 the roots must sum to 5 and multiply to 6, which 2 and 3 satisfy.
Is there any difference between factoring and the quadratic formula?
Both give the same roots. Factoring is faster when the roots are whole numbers or simple fractions, while the quadratic formula is the guaranteed method that always works. Whenever an equation resists factoring, switch to the formula (or to this calculator).