Solving Cubic Equations with the Cardano Formula: A Detailed Guide
This article discusses the process of solving a specific cubic equation using the Cardano formula. We'll start with the given equation and proceed step-by-step to find the solutions, providing a detailed explanation of the steps involved, including graphing and using complex numbers.
Solving a Cubic Equation
Consider the given equation:
x^3 - x^2 - 3x x - 1
To simplify, we can rewrite the equation as:
x^3 - 3x - 1 0
This is a cubic equation of the form x^3 px q 0, where:
p -3 q -1Using the Cardano Formula
The Cardano formula is a method for solving cubic equations of the form x^3 px q 0. The formula is given by:
x sqrt[3]{-frac{q}{2} sqrt{frac{q^2}{4} frac{p^3}{27}}} sqrt[3]{-frac{q}{2} - sqrt{frac{q^2}{4} frac{p^3}{27}}}
Substitute p -3 and q -1 into the formula:
x sqrt[3]{frac{1}{2} sqrt{frac{1}{4} - frac{9}{27}}} sqrt[3]{frac{1}{2} - sqrt{frac{1}{4} - frac{9}{27}}}
Simplify the terms inside the square root:
x sqrt[3]{frac{1}{2} sqrt{frac{1}{4} - frac{1}{3}}} sqrt[3]{frac{1}{2} - sqrt{frac{1}{4} - frac{1}{3}}}
Since frac{1}{4} - frac{1}{3} frac{3}{12} - frac{4}{12} -frac{1}{12}
x sqrt[3]{frac{1}{2} sqrt{-frac{1}{12}}} sqrt[3]{frac{1}{2} - sqrt{-frac{1}{12}}}
This results in complex numbers due to the square root of a negative term:
x sqrt[3]{frac{1}{2} cdot frac{sqrt{3}}{2}i} sqrt[3]{frac{1}{2} - frac{sqrt{3}}{2}i}
Express the complex numbers in the form of costheta isintheta:
x sqrt[3]{cosfrac{2pi}{3} sinfrac{2pi}{3}i} sqrt[3]{cosfrac{-2pi}{3} sinfrac{-2pi}{3}i}
To simplify, we can use the exponential form of complex numbers:
x e^{frac{2pi i}{3}} cdot sqrt[3]{e^{frac{-2pi i}{3}}} e^{frac{-2pi i}{3}} cdot sqrt[3]{e^{frac{2pi i}{3}}}
(2pi i), and we need to find the appropriate values of 2kpi /9 and 2lpi /9 for the roots:
(-frac{3kpi}{9} frac{2lpi}{9} -frac{3kpi}{9} frac{2lpi}{9} -frac{pi}{3}
The solutions are:
(x -1.8794 (x 0.3473 (x 1.5321Graphical Interpretation
To verify the solutions, let's consider the graph of the function y x^3 - 3x - 1 - (x - 1). The graph will show three real roots that correspond to the values we found.
Using graphing tools like Desmos, we observe three real roots in the range [-2, 2], which match our solutions.
Conclusion
In this article, we used the Cardano formula to solve the given cubic equation and verified the solutions graphically. Understanding the steps involved, from simplifying the equation to using complex numbers and the exponential form, is key to solving such problems accurately.