What does a negative discriminant mean?

A negative discriminant (D < 0) means the quadratic equation has no real roots. Instead, it has two complex roots containing the imaginary unit i (where i² = -1).

Why can't coefficient "a" be zero?

If a = 0, the equation reduces to bx + c = 0, which is a linear equation, not a quadratic equation.

What is the vertex of a parabola?

The vertex is the minimum or maximum point of the parabola, located at x = -b / 2a.

Math Tool✨ New Release

Quadratic Equation Calculator

Solve quadratic equations of the form ax² + bx + c = 0. Supports real and complex imaginary roots, with full step-by-step discriminant details.

Interactive Quadratic Equation Calculator Workspace

Live Solver

Coefficient a (x²)

Coefficient b (x)

Constant c

Calculated Roots

x₁ = 1.5

x₂ = 1

Step-by-Step Roots Breakdown

We want to solve: 2x² + (-5)x + (3) = 0

Step 1: Compute discriminant D = b² - 4ac

D = (-5)² - 4 × (2) × (3)

D = 25 - 24 = 1

D > 0, so there are two distinct real roots.

Step 2: Apply Quadratic Formula: x = (-b ± √D) / 2a

x₁ = (-(-5) + √1) / (2 × 2) = (5 + 1.0000) / 4 = 1.5000

x₂ = (-(-5) - √1) / (2 × 2) = (5 - 1.0000) / 4 = 1.0000

Mathematical Formula & Variables

Equation Modelx = (-b ± √(b² - 4ac)) / 2a

The quadratic formula finds the solutions (roots) of any quadratic equation. The term b² - 4ac is called the discriminant (D) and determines the nature of the roots.

Variable Definitions

Symbol	Description
a	Coefficient of x² (must not be zero)
b	Coefficient of x
c	Constant numerical term
x	Calculated solutions/roots of the equation

How to Use the Quadratic Equation Calculator

Input the coefficients a, b, and c into the quadratic equation form.
Ensure that coefficient 'a' is not zero.
Click calculate to instantly see the real or imaginary roots, along with the step-by-step discriminant analysis.