To solve a cubic equation using **Cardano’s Formula**, first express the cubic equation in the general form. Identify the coefficients and calculate the values for **g** and **f**. Then, plug these values into the Cardano’s formula to compute the root of the equation.

## Cardano’s Formula Calculator

**Cardano’s Formula** is used to solve cubic equations of the form $ax^3 + bx^2 + cx + d = 0$. It is a crucial method in algebra for finding the roots of cubic equations. The formula simplifies the complex process of finding the roots of a cubic equation.

**Formula:**

Here’s the general formula:

$x = \left(3 – 2g + 4g^2 + 27f^3\right)^{\frac{1}{3}} + \left(3 – 2g – 4g^2 + 27f^3\right)^{\frac{1}{3}} – \frac{3ab}{3}$

Where:

**x**is the variable we are solving for,**g**and**f**are derived from the original cubic equation,**a**and**b**are coefficients from the cubic equation.

**How to Calculate ?**

- Rewrite the cubic equation in the general form.
- Identify the coefficients and calculate
**g**and**f**from the cubic equation. - Substitute these values into
**Cardano’s formula**to compute the root(s).

**Example 1:**

Calculation |
Instructions |
---|---|

Given: $x^3 – 3x + 2 = 0$ |
This is a cubic equation in general form. |

Identify $g$ and $f$ values. | Use the coefficients to calculate $g = -3$ and $f = 2$. |

Apply Cardano’s Formula: | $x = \left(3 – 2(-3) + 4(-3)^2 + 27(2)^3\right)^{\frac{1}{3}} + \left(3 – 2(-3) – 4(-3)^2 + 27(2)^3\right)^{\frac{1}{3}} – \frac{3ab}{3}$ |

Simplified: $x = 2$ | The root of the cubic equation is $x = 2$. |

**Answer:** The root of the cubic equation is **x = 2**.

**Example 2:**

Calculation |
Instructions |
---|---|

Given: $x^3 + 6x^2 + 11x + 6 = 0$ |
This is another cubic equation in general form. |

Identify $g$ and $f$ values. | Calculate $g = 6$ and $f = 11$. |

Apply Cardano’s Formula: | $x = \left(3 – 2(6) + 4(6)^2 + 27(11)^3\right)^{\frac{1}{3}} + \left(3 – 2(6) – 4(6)^2 + 27(11)^3\right)^{\frac{1}{3}} – \frac{3ab}{3}$ |

Simplified: $x = -1$ | The root of the cubic equation is $x = -1$. |

**Answer:** The root of the cubic equation is **x = -1**.

**What is Cardano’s Formula Calculator ?**

The **Cardano’s Formula Calculator** is a valuable tool for solving cubic equations, which are polynomials of the form $ax^3 + bx^2 + cx + d = 0$. Named after the Italian mathematician Gerolamo Cardano, this formula provides a method for finding the roots of cubic equations, whether they have one real root or three real roots.

To use the calculator, you will typically need to input the coefficients of your cubic equation. The calculator will then apply Cardano’s method to find the roots. This involves several steps, including transforming the cubic equation into a depressed form (eliminating the quadratic term), and then using the cubic formula to determine the roots.

You may be wondering, **how to use Cardano’s formula?** First, rewrite your cubic equation in the standard form. Then, identify the coefficients $a$, $b$, $c$, and $d$. The calculator will guide you through the calculations to find the roots based on Cardano’s method.

Understanding **what Cardano’s formula is** can also be helpful. It’s a solution for cubic equations derived from the relationships between the roots and coefficients. The formula provides a systematic way to find solutions using radicals, even when complex numbers are involved.

If you’re looking for specific examples, you can enter equations like $x^3 – 3x – 10 = 0$ into the calculator. The tool will illustrate the process step by step, allowing you to see how Cardano’s method is applied to obtain the roots.

**Final Words:**

In summary, the Cardano’s Formula Calculator is an essential resource for students and professionals working with cubic equations. It simplifies the process of finding roots and enhances your understanding of polynomial functions and their behaviors. Whether you’re studying for an exam or tackling complex mathematical problems, this calculator will be a valuable aid.