# Cubic Equation Solver

Solve cubic equations, classify the root pattern, and show Cardano-style intermediate values

> Canonical page: https://elysiatools.com/en/tools/cubic-equation-solver

- **Category:** Math & Numbers

- **Keywords:** cubic equation, cardano, polynomial roots, discriminant, math

## Overview

The Cubic Equation Solver is a specialized mathematical utility designed to calculate the roots of third-degree polynomials. Whether you input individual coefficients or a complete equation string, it instantly computes real and complex roots, classifies the root pattern based on the discriminant, and provides Cardano-style intermediate values for deeper mathematical analysis.

## Inputs

- **Input Format** (select)
- **Coefficient a / Equation** (text): 1 or x^3 - 6x^2 + 11x - 6 = 0
- **Coefficient b** (number)
- **Coefficient c** (number)
- **Coefficient d** (number)
- **Decimal Places** (number)
- **Show Steps** (checkbox)

## When to use

- When you need to find the exact real and complex roots of a third-degree polynomial equation.
- When verifying manual calculations using Cardano's formula for academic or engineering purposes.
- When analyzing the discriminant to classify the root pattern of a cubic function.

## How it works

- Select your preferred input format: either individual coefficients (a, b, c, d) or a complete equation string.
- Enter the equation or coefficients into the designated input fields.
- Adjust the decimal places for precision and toggle the option to show intermediate calculation steps.
- View the structured JSON output containing the calculated roots, discriminant, and Cardano-style intermediate values.

## Use cases

- Engineering students verifying manual root calculations for control systems or fluid dynamics assignments.
- Mathematics educators generating step-by-step examples of Cardano's method for classroom demonstrations.
- Software developers needing a quick way to validate custom polynomial root-finding algorithms.

## Frequently asked questions

### Can I input the full equation instead of just coefficients?

Yes, you can switch the Input Format to 'Complete Equation' and type the full string, such as x^3 - 6x^2 + 11x - 6 = 0.

### Does this tool calculate complex roots?

Yes, the solver calculates and returns both real and complex roots based on the equation's discriminant.

### What are Cardano-style intermediate values?

These are the intermediate variables (often denoted as p, q, and the discriminant) used in Cardano's formula to solve cubic equations analytically.

### Can I control the precision of the output?

Yes, you can adjust the 'Decimal Places' setting between 0 and 10 to round the calculated roots to your required precision.

### What happens if the coefficient 'a' is zero?

If 'a' is zero, the equation is quadratic, not cubic. You must provide a non-zero value for the 'a' coefficient for a valid cubic equation.

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