What is the discriminant?

It's the value b² - 4ac. If it's positive, you have two real roots. If zero, one real root. If negative, the roots are complex (imaginary).

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Mathematics

Quadratic Equation Solver – Find Roots Instantly | CalcTools

Solve quadratic equations of the form ax² + bx + c = 0. Find real and complex roots with step-by-step logic.

Quadratic equations are the foundation of motion and optimization. From calculating the trajectory of a ball to finding the most efficient dimensions for a physical object, the ability to solve for 'x' in a second-degree polynomial is a fundamental skill in physics and engineering. Our Quadratic Solver is a specialized math service that finds both real and complex roots instantly. It eliminates the tedious manual labor of the quadratic formula, allowing you to focus on the application of the results rather than the mechanics of the calculation.

Resulting Roots

x₁ = -2.0000, x₂ = -3.0000

Roots for 1x² + 5x + 6 = 0 are -2.0000 and -3.0000

x = [-b ± √(b² - 4ac)] / 2a

Discriminant (Δ)1

How it Works

The solver first computes the discriminant to determine the nature of the roots. It then applies the quadratic formula to find the exact values, correctly handling the real and imaginary components when the solution requires complex numbers.

Calculation Formula

x = [-b ± √(b² - 4ac)] / 2a

Calculation Examples

1. **Real**: x² - 5x + 6 = 0 roots are **2 and 3**. 2. **Complex**: x² + 1 = 0 roots are **±i**.

Why use this tool in your daily life?

This tool is an essential productivity booster for students and researchers. By providing instant, error-free roots for any coefficient, it serves as a reliable validator for your homework and a high-speed tool for real-world modeling.