Problem 31 - A History of Mathematics: An Introduction

Question

Use Cardano's formula to solve \(x^{3}=6 x+6\)

Step-by-Step Solution

Answer

Answer: The real root of the given equation according to Cardano's formula is approximately \(x \approx 0.411\).

Step 1: Rewrite the equation in the form of a depressed cubic

In general, a cubic equation can be written as \(ax^3 + bx^2 + cx + d = 0\). A depressed cubic equation is an equation without a quadratic term, meaning it is in the form of \(x^3 + px + q = 0\). To rewrite the given equation \(x^{3}=6x+6\) in the form of a depressed cubic, we can directly observe that \(p = 6\) and \(q = 6\); the equation can therefore be written as \(x^3 + 6x + 6 = 0\).

Step 2: Calculate Cardano's formula parameters

Cardano's formula involves two parameters, \(\Delta\) and \(u\), which can be calculated using the values of \(p\) and \(q\). They are given by the following formulas: \(\Delta = -\frac{27q^2}{64} + \frac{16p^3}{27}\) and \(u = \sqrt[3]{\frac{-27q+9\sqrt\Delta}{54}}\) Here, \(p = 6\) and \(q = 6\). Let's calculate the values of \(\Delta\) and \(u\): \(\Delta = -\frac{27(6)^2}{64} + \frac{16(6)^3}{27} = -\frac{1458}{64} + \frac{13824}{27} = -22.78125 + 512 = 489.21875\) \(u = \sqrt[3]{\frac{-27(6)+9\sqrt{489.21875}}{54}} = \sqrt[3]{\frac{-162+9\sqrt{489.21875}}{54}} \approx 1.638\)

Step 3: Calculate the roots using Cardano's formula

Cardano's formula calculates a real root of a depressed cubic equation using the parameters \(\Delta\) and \(u\). The formula is given by: \(x = u - \frac{p}{3u}\) From Step 2, we have \(p = 6\) and \(u \approx 1.638\). Let's calculate the real root using Cardano's formula: \(x = 1.638 - \frac{6}{3(1.638)} \approx 1.638 - 1.227 \approx 0.411\)

Step 4: Conclusion

Using Cardano's formula, we found that the real root of the given equation \(x^3=6x+6\) is approximately \(x \approx 0.411\). Note that the equation may have further complex roots, but Cardano's formula is sufficient for finding the real root.

Key Concepts

Depressed Cubic Equation

A cubic equation is a polynomial equation of degree three typically represented in the form: \(ax^3 + bx^2 + cx + d = 0\). It includes a squared term, a linear term, and a constant. However, in some special cases, the quadratic term can be eliminated, resulting in a simpler form known as the depressed cubic equation. This special form is expressed as \(x^3 + px + q = 0\), where the quadratic coefficient is zero. This simplification offers considerable computational convenience, particularly when solving for roots. Many methods, like Cardano's formula, work exclusively with this form because it reduces calculation complexity and introduces fewer terms to manage.

Cubic Equation

Cubic equations are fundamental in algebra and represent polynomial equations of the third degree. The general form, as mentioned, is \(ax^3 + bx^2 + cx + d = 0\), where \(a, b, c,\) and \(d\) are constants. These equations can describe various complex phenomena in physics and engineering.
Solving cubic equations involves finding values of \(x\) that satisfy the equation. These values could be real or complex numbers. A critical insight about cubic equations is that they always have three roots, although not all of them need to be real. The coefficients' values directly influence the nature (real or complex) of the roots. Various mathematical tools, including graphical methods and algebraic formulas, are used to find these roots.

Real Root Calculation

Real root calculation in cubic equations is essential for practical applications where real solutions are required, such as physics or engineering problems. Cardano's formula, named after the Italian mathematician Gerolamo Cardano, provides a way to find at least one real root of a depressed cubic equation.
The formula works by calculating two parameters, \(\Delta\) and \(u\). These are derived from the coefficients \(p\) and \(q\) of the equation \(x^3 + px + q = 0\).

• First, calculate \(\Delta = -\frac{27q^2}{64} + \frac{16p^3}{27}\).
• Then, using \(\Delta\), \(u\) is calculated as \(\sqrt[3]{\frac{-27q+9\sqrt\Delta}{54}}\).

Using these values, the real root is found with the expression \(x = u - \frac{p}{3u}\). This method focuses on determining a single real root, which can be crucial for cases where an exact real solution is needed.

Mathematical Solution Methods

Various mathematical solution methods have been developed over centuries to solve cubic equations. While Cardano's formula is one classical approach, other techniques are helpful depending on the equation's nature, including:

• Graphical Methods: Plotting the equation to visually identify real roots.
• Numerical Methods: Such as the Newton-Raphson method for approximating roots.
• Algebraic Methods: Utilizing algebraic identities and transformations to simplify the equation.
• Complex Analysis: Leveraging the power of complex numbers to find all roots effectively.

Each method offers distinct advantages and efficiencies based on the problem's specific requirements. Understanding these methods provides a more comprehensive toolkit for tackling polynomial equations across different scenarios.