Solving The Quartic Equation 4x⁴ - 4x² = 8 A Step-by-Step Guide
Introduction: Delving into Quartic Equations
In the realm of mathematics, solving equations is a fundamental pursuit. Among the various types of equations, quartic equations, which are polynomial equations of the fourth degree, hold a significant place. This article aims to provide a comprehensive exploration of the solutions to the specific quartic equation 4x⁴ - 4x² = 8. We will embark on a step-by-step journey, employing algebraic techniques to unravel the roots of this equation. Our primary focus is to transform the equation into a more manageable form, ultimately leading us to the desired solutions. The solutions will be determined through careful algebraic manipulation and analysis, providing a clear and understandable path to the final answer. This exploration will not only reveal the solutions but also deepen our understanding of quartic equations and their inherent properties. Understanding how to solve quartic equations is crucial in various fields, including engineering, physics, and computer science, where these equations often arise in modeling complex systems and phenomena. The systematic approach we will adopt here can be applied to a broader range of quartic equations, making this a valuable learning experience for anyone interested in mathematics and its applications. By the end of this article, you will not only know the solutions to this specific equation but also have a better grasp of the techniques involved in solving quartic equations in general. This knowledge will empower you to tackle similar problems with confidence and precision.
Transforming the Equation: A Step-by-Step Approach
To effectively solve the quartic equation 4x⁴ - 4x² = 8, we initiate a strategic transformation process. The first step involves simplifying the equation by dividing both sides by the common factor of 4. This crucial step not only reduces the complexity of the coefficients but also sets the stage for subsequent manipulations. The transformed equation, x⁴ - x² = 2, is now more amenable to further analysis. Our next objective is to bring all terms to one side, resulting in the standard form of a polynomial equation. Subtracting 2 from both sides, we arrive at the equation x⁴ - x² - 2 = 0. This form is particularly significant because it resembles a quadratic equation, a type of equation we are well-equipped to solve. To capitalize on this resemblance, we introduce a substitution technique. Let y = x². This substitution transforms the quartic equation into a quadratic equation in terms of y. The equation x⁴ - x² - 2 = 0 now becomes y² - y - 2 = 0. This quadratic equation is significantly easier to handle than the original quartic equation. We can now apply standard techniques for solving quadratic equations, such as factoring or using the quadratic formula. The transformation process we have undertaken highlights the power of algebraic manipulation in simplifying complex equations. By strategically transforming the original equation, we have successfully converted it into a more familiar and manageable form. This approach is a cornerstone of mathematical problem-solving and demonstrates the elegance of using substitutions to simplify equations. The next step involves solving this quadratic equation for y, which will then lead us back to the solutions for x.
Solving the Quadratic: Unveiling the Roots
Having successfully transformed the quartic equation into the quadratic equation y² - y - 2 = 0, our focus now shifts to solving this quadratic equation. There are several methods available for this purpose, but factoring is often the most efficient when applicable. We seek two numbers that multiply to -2 and add up to -1. These numbers are -2 and 1. Thus, we can factor the quadratic equation as (y - 2)(y + 1) = 0. This factored form immediately reveals the roots of the quadratic equation. Setting each factor equal to zero, we obtain two possible solutions for y: y - 2 = 0 or y + 1 = 0. Solving these linear equations, we find y = 2 and y = -1. These values of y represent the solutions to the quadratic equation, but they are not the solutions to our original quartic equation. Recall that we made the substitution y = x². We must now reverse this substitution to find the values of x. This involves substituting the values of y back into the equation y = x². For y = 2, we have x² = 2. Taking the square root of both sides, we obtain x = ±√2. These are two of the solutions to the quartic equation. For y = -1, we have x² = -1. Taking the square root of both sides, we obtain x = ±√(-1), which simplifies to x = ±i, where i is the imaginary unit. These are the remaining two solutions to the quartic equation. The process of solving the quadratic equation and then reversing the substitution demonstrates the interconnectedness of different mathematical concepts. By leveraging our knowledge of quadratic equations, we were able to efficiently solve a more complex quartic equation. This highlights the importance of mastering fundamental algebraic techniques, as they often serve as building blocks for solving more advanced problems. The solutions we have found, x = ±√2 and x = ±i, represent the complete set of roots for the given quartic equation.
Identifying the Solutions: A Comprehensive Overview
Having navigated the steps of transformation and solution, we have successfully unearthed the solutions to the quartic equation 4x⁴ - 4x² = 8. Our journey began with simplifying the equation, followed by a strategic substitution that converted the quartic into a more manageable quadratic form. Solving the quadratic equation yielded two values for y, which we then used to find the corresponding values for x. The solutions we have identified are x = ±√2 and x = ±i. These four values represent the complete set of roots for the given quartic equation. It is crucial to recognize that quartic equations, being polynomial equations of the fourth degree, can have up to four solutions. In this case, we have found all four solutions, two of which are real (x = ±√2) and two of which are imaginary (x = ±i). The presence of both real and imaginary solutions is a common characteristic of quartic equations and highlights the richness and complexity of polynomial equations. Understanding the nature of these solutions is essential in various mathematical and scientific contexts. Real solutions correspond to points where the graph of the polynomial intersects the x-axis, while imaginary solutions do not have a direct graphical interpretation in the real plane. The ability to identify and interpret both real and imaginary solutions is a testament to a strong understanding of algebraic principles. The process we have undertaken in solving this equation serves as a valuable template for tackling other quartic equations. By mastering the techniques of simplification, substitution, and quadratic equation solving, you can confidently approach a wide range of polynomial equations. The solutions x = ±√2 and x = ±i provide a definitive answer to the original problem and underscore the power of algebraic methods in unraveling mathematical mysteries.
Conclusion: A Mathematical Triumph
In conclusion, we have successfully navigated the intricate path to solving the quartic equation 4x⁴ - 4x² = 8. Through a series of strategic algebraic manipulations, we transformed the equation into a manageable form, solved the resulting quadratic equation, and ultimately identified the complete set of solutions. The solutions we have found are x = ±√2 and x = ±i. These values represent the roots of the quartic equation and demonstrate the power and elegance of algebraic techniques. The process we undertook involved several key steps, each of which played a crucial role in arriving at the final answer. First, we simplified the equation by dividing both sides by a common factor. This step reduced the complexity of the coefficients and paved the way for subsequent manipulations. Next, we employed a substitution technique, replacing x² with y, which transformed the quartic equation into a quadratic equation. This transformation was a pivotal moment in our solution process, as it allowed us to leverage our knowledge of quadratic equations to solve a more complex problem. We then solved the quadratic equation using factoring, a standard technique for finding the roots of quadratic equations. The solutions for y were then used to find the corresponding solutions for x, which involved reversing the substitution we had made earlier. This final step yielded the four solutions to the quartic equation, two real and two imaginary. The journey of solving this equation has not only provided us with the solutions but has also deepened our understanding of quartic equations and the techniques used to solve them. The systematic approach we adopted can be applied to a broader range of polynomial equations, making this a valuable learning experience. The solutions x = ±√2 and x = ±i stand as a testament to our mathematical prowess and underscore the beauty and power of algebraic methods in solving complex problems.
Therefore, the correct answers are C. x = ± √2 and A. x = ± i.
