What if the quadratic coefficient a is 0?

If a = 0, the equation reduces to a linear equation bx + c = 0 with single solution x = -c / b.

Quadratic Equation Calculator : Solve ax² + bx + c = 0 with Discriminant

Q: What happens when the discriminant is negative?

When Δ < 0, the equation has no real roots and instead produces two complex conjugate solutions with imaginary component i = √(-1).

The Quadratic Equation Calculator finds the roots of second-degree polynomial equations ax² + bx + c = 0. It calculates the discriminant (Δ = b² - 4ac), identifies vertex coordinates, and outputs exact real or complex roots.

Interactive Calculator

100% Client-Side Engine

Quadratic Roots (two_real)

x₁ = 3, x₂ = 2

Discriminant (D = b² - 4ac)

Parabola Vertex (h, k)

(2.5, -0.25)

Parabola Orientation

Opens upward

Quadratic Formula Derivation

1. Standard Form: 1x² + -5x + 6 = 0

2. Discriminant: D = b² - 4ac = (-5)² - 4(1)(6) = 1

3. D > 0: Two distinct real roots:

x₁ = (-b + √D) / 2a = (5 + 1) / 2 = 3

x₂ = (-b - √D) / 2a = (5 - 1) / 2 = 2

4. Vertex: (2.5, -0.25)

Step-by-Step Examples

Solving x² - 5x + 6 = 0

Factoring quadratic with two distinct real roots.

Inputs: a: 1, b: -5, c: 6

Result: x₁ = 3, x₂ = 2 (Δ = 1 > 0)

Δ = (-5)² - 4(1)(6) = 25 - 24 = 1. Roots are (5 ± 1)/2 = 3 and 2.

How to Use This Tool

Enter coefficient a (quadratic term).
Enter coefficient b (linear term) and constant c.
Click Calculate to see the discriminant, nature of roots, vertex (h, k), and exact roots.

Frequently Asked Questions

When Δ < 0, the equation has no real roots and instead produces two complex conjugate solutions with imaginary component i = √(-1).

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