Solving the Equation x4 - 2x2 24: A Quadratic Substitution Approach
The equation (x^4 - 2x^2 24) is a polynomial equation that can be approached in several ways. One effective method is to utilize a substitution technique that simplifies the problem into a more manageable form. This article will guide you through the process of solving this equation using a quadratic substitution.
Introduction to the Problem
Let's start by understanding the equation we are dealing with: x^4 - 2x^2 24. This equation may seem complex at first glance, but it can be transformed into a more familiar form through a suitable substitution.
Quadratic Substitution
The key to solving this equation lies in recognizing that it is a polynomial in terms of (x^2). We can make a substitution (u x^2), which simplifies the equation significantly. By substituting (u x^2), the equation becomes:
u^2 - 2u 24
This is a quadratic equation in terms of (u), which we can solve using standard methods.
Step-by-Step Solution
Let's solve the quadratic equation u^2 - 2u - 24 0 step-by-step.
Standard Form of the Quadratic Equation
First, we rewrite the equation in standard form:
u^2 - 2u - 24 0
Finding the Roots
We can solve this quadratic equation using the quadratic formula (u frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a 1), (b -2), and (c -24).
Plugging in the values, we get:
u frac{-(-2) pm sqrt{(-2)^2 - 4 cdot 1 cdot (-24)}}{2 cdot 1}
u frac{2 pm sqrt{4 96}}{2}
u frac{2 pm sqrt{100}}{2}
u frac{2 pm 10}{2}
Therefore, the solutions are:
u frac{12}{2} 6 quad text{or} quad u frac{-8}{2} -4
Hence, (u 6) or (u -4).
Back-Substitution
Now that we have solved for (u), we need to substitute (x^2) back into the equation.
From the substitution (u x^2), we have:
x^2 6 quad text{or} quad x^2 -4
For (x^2 6), we take the square root of both sides:
x pm sqrt{6}
For (x^2 -4), we recognize that this involves complex numbers:
x pm sqrt{-4} pm 2i
Conclusion
Therefore, the solutions to the equation (x^4 - 2x^2 24) are:
x pm sqrt{6} quad text{or} quad x pm 2i
These solutions showcase the power of quadratic substitution in solving higher-order polynomial equations.
Related Keywords
Quadratic substitution Solving equations Complex solutionsConclusion
The article has demonstrated the step-by-step process of solving the polynomial equation (x^4 - 2x^2 24) using a quadratic substitution. This method simplifies the problem into a more familiar quadratic form, making it easier to find the roots. By understanding and applying quadratic substitution, you can tackle a wide range of polynomial equations with confidence.