Solving Cubic Equations using Cardan’s Method: A Step-by-Step Guide
Introduction to Cardan’s Method
Cardan’s method is a powerful technique used to find the roots of a cubic equation. This method involves transforming the given cubic equation into a simpler form and then solving for the roots through a series of algebraic manipulations. This article will walk you through the steps of solving the cubic equation x^3 - x^2 - 16x 20 0 using Cardan’s method.
Steps to Solve the Cubic Equation Using Cardan’s Method
Step 1: Transform the Cubic Equation
First, we need to transform the given cubic equation to eliminate the second-degree term. We do this by expressing x as y a for some constant a. Starting with the equation:
x^3 - x^2 - 16x 20 0
Setting x y - a, we get:
(y - a)^3 - (y - a)^2 - 16(y - a) 20 0
Expanding and simplifying the equation:
y^3 - 3ay^2 3a^2y - a^3 - y^2 2ay - a^2 - 16y 16a 20 0
Combining like terms:
y^3 - (3a 1)y^2 (3a^2 2a - 16)y - (a^3 a^2 - 16a 20) 0
For the equation to be in the form y^3 by^2 cy d 0, we need:
- (3a 1) 0
Solving for a:
a -1/3
Step 2: Substitute the Value of a
Substituting a -1/3 into the equation:
y^3 - (3(-1/3) 1)y^2 - (3(-1/3)^2 2(-1/3) - 16)y - ((-1/3)^3 (-1/3)^2 - 16(-1/3) 20) 0
Simplifying:
y^3 - 0y^2 - (1/3 - 2/3 - 16)y - (1/27 - 1/9 16/3 20) 0
y^3 - (-14/3)y - (686/27) 0
Multiplying by 27 to clear the denominators:
27y^3 - 1473y - 686 0
Step 3: Apply Cardan’s Formula
To solve the transformed equation using Cardan’s method, we express 3y u v where uv -b/3. Here, b -1473/27
Choosing u and v such that:
u v 49
uv -49/3
Solving for u and v:
u^2 - 49u - 343/3 0
Solving the quadratic equation:
u^2 - 49u - 343/3 0
Using the quadratic formula:
u 7
v -7
Thus, 3y 7 - 7 14
y -14/3
Step 4: Find the Roots of the Original Equation
Now, we substitute y -14/3 back into x y - a:
x -14/3 1/3 -5
So, one root of the original equation is x -5.
By factoring the original cubic equation:
x^3 - x^2 - 16x 20 0
Using the factor theorem, we find that x 5 is a factor. Using synthetic division:
x^3 - x^2 - 16x 20 (x 5)(x^2 - 6x 4)
Factoring the quadratic equation:
x^2 - 6x 4 (x - 2)(x - 2)
Therefore, the roots of the original equation are:
x -5, x 2 (double root)
Conclusion
In this article, we have demonstrated how to solve a cubic equation using Cardan’s method. While this method can be lengthy, it provides a systematic approach to finding the roots of a cubic equation. This technique is particularly useful for those who are interested in delving deeper into algebraic solvability techniques.