Roots of Symbolic Cubic Equation Calculator

How to Use This Calculator

To find the roots of a cubic equation:

1. Enter the coefficients a, b, c, and d in the calculator form.
2. Click "Calculate Roots" to solve the equation.
3. Review the exact solutions or numerical approximations provided.
4. Use the chart to visualize the roots on the complex plane.

The calculator handles both real and complex roots, providing exact solutions when possible and numerical approximations otherwise.

Mathematical Background

A general cubic equation has the form:

where a, b, c, and d are real numbers and a ≠ 0. The roots of the equation are the values of x that satisfy it.

Cardano's Formula

The roots can be found using Cardano's formula, which provides exact solutions when the discriminant Δ is non-zero:

When Δ > 0, there are three distinct real roots. When Δ = 0, there are multiple roots. When Δ < 0, there is one real root and two complex conjugate roots.

Worked Example

Let's solve the equation x³ - 6x² + 11x - 6 = 0.

1. Identify the coefficients: a = 1, b = -6, c = 11, d = -6.
2. Calculate the discriminant: Δ = 18(1)(-6)(11)(-6) - 4(-6)³(-6) + (-6)²(11)² - 4(1)(11)³ - 27(1)²(-6)² = 3888 - 864 + 4356 - 5796 - 972 = -1196.
3. Since Δ < 0, there is one real root and two complex conjugate roots.
4. The real root is x = 1, and the complex roots are x = 2 ± √3i.

This example demonstrates how the calculator can handle both real and complex roots.

Interpreting the Results

The calculator provides:

• Exact solutions when possible.
• Numerical approximations for complex roots.
• A visualization of the roots on the complex plane.

Understanding the nature of the roots (real or complex) is crucial for applications in engineering and physics.