Solving X⁴ - 24 = 0 A Step-by-Step Guide

Introduction to Solving Quartic Equations

In the realm of algebra, solving equations is a fundamental skill, and among the various types of equations, quartic equations hold a significant place due to their complexity and the methods required to tackle them. Quartic equations, which are polynomial equations of the fourth degree, can initially seem daunting, but with the right approach, they can be systematically solved. This guide will focus on solving the specific quartic equation X⁴ - 24 = 0 using a step-by-step methodology. Before diving into the solution, it's beneficial to understand the nature of quartic equations and the general strategies employed to find their roots. Quartic equations, in their general form, are expressed as ax⁴ + bx³ + cx² + dx + e = 0, where a, b, c, d, and e are constants and a ≠ 0. The roots of a quartic equation are the values of x that satisfy the equation, and these can be real or complex numbers. Unlike quadratic equations, which have a straightforward formula (the quadratic formula) for finding roots, quartic equations require more sophisticated techniques. These techniques often involve factoring, substitution, and, in some cases, the use of complex numbers. In this guide, we'll explore a method that simplifies the given equation, X⁴ - 24 = 0, allowing us to find its solutions effectively. By understanding the underlying principles and applying the appropriate steps, you'll gain confidence in tackling similar algebraic problems. Let's embark on this mathematical journey and unravel the solutions to this interesting quartic equation.

Step 1: Isolating the Variable Term

To effectively begin solving the equation X⁴ - 24 = 0, the first crucial step involves isolating the variable term. This process sets the stage for subsequent operations that will lead us closer to finding the roots of the equation. Isolating the variable term means manipulating the equation to get the term containing X⁴ by itself on one side of the equation. In this case, we have X⁴ - 24 = 0. To isolate X⁴, we need to eliminate the constant term, which is -24. This can be achieved by performing the inverse operation on both sides of the equation. The inverse operation of subtraction is addition, so we add 24 to both sides of the equation. This maintains the equality and moves us closer to our goal. Adding 24 to both sides gives us: X⁴ - 24 + 24 = 0 + 24. Simplifying this, we get X⁴ = 24. Now, the variable term X⁴ is isolated on one side of the equation, making it easier to proceed with the next steps. This isolation is a critical foundation for further solving, as it allows us to focus solely on the term containing the variable and its relationship to the constant on the other side. This step is not only essential for this particular equation but also a common strategy in solving various algebraic equations. By isolating the variable term, we simplify the equation and pave the way for applying other techniques, such as taking roots or factoring, to find the solutions. With X⁴ now isolated, we are well-positioned to tackle the remaining steps in finding the values of X that satisfy the equation.

Step 2: Taking the Fourth Root

Having successfully isolated the variable term in the equation X⁴ = 24, the next logical step in solving for X is to take the fourth root of both sides. This operation is the inverse of raising to the fourth power and will help us reduce the exponent on X to 1, effectively revealing the possible values of X. When dealing with even roots, such as the fourth root, it's essential to remember that there will be both positive and negative real solutions, as well as complex solutions. This is because raising either a positive or a negative number to an even power results in a positive number. To take the fourth root of both sides of the equation X⁴ = 24, we apply the fourth root operator: ⁴√(X⁴) = ⁴√24. The fourth root of X⁴ is simply X, so the left side of the equation becomes X. The right side, ⁴√24, represents the fourth root of 24. To simplify ⁴√24, we can look for factors of 24 that are perfect fourth powers. However, 24 does not have any perfect fourth power factors other than 1. Therefore, we can express ⁴√24 in simplest radical form by factoring 24 into its prime factors: 24 = 2³ × 3. Thus, ⁴√24 = ⁴√(2³ × 3). We can rewrite this as ⁴√24 = ⁴√2³ * ⁴√3. Since we cannot simplify this further without resorting to approximations or complex numbers, we leave it in this form for now. It's crucial to remember that when taking the fourth root, we must consider both the positive and negative roots. Therefore, X = ± ⁴√24. This indicates that we have two real solutions: X = ⁴√24 and X = -⁴√24. Additionally, there are two complex solutions, which we will explore in the next step. This step of taking the fourth root is pivotal in solving quartic equations of this form, as it directly leads us to the potential values of the variable. By understanding the properties of roots and considering both positive and negative solutions, we ensure a comprehensive approach to finding all possible solutions.

Step 3: Finding the Real Solutions

In the process of solving the equation X⁴ - 24 = 0, we've reached a point where we've isolated X and have the expression X = ± ⁴√24. This step focuses on identifying the real solutions that arise from this expression. Real solutions are the values of X that are real numbers, meaning they can be plotted on a number line. As we've already established, the expression ± ⁴√24 indicates that we have two potential real solutions: one positive and one negative. The positive real solution is X = ⁴√24, which represents the principal fourth root of 24. This is the positive real number that, when raised to the fourth power, equals 24. To get a sense of the numerical value of ⁴√24, we can estimate it or use a calculator. Since 2⁴ = 16 and 3⁴ = 81, the fourth root of 24 lies between 2 and 3. A calculator can provide a more precise approximation, which is approximately 2.213. The negative real solution is X = -⁴√24, which is the negative of the principal fourth root of 24. This is the negative real number that, when raised to the fourth power, also equals 24. Its approximate numerical value is -2.213. These two real solutions are significant because they are the points where the graph of the function Y = X⁴ - 24 intersects the X-axis. In other words, they are the X-values for which the function equals zero. When solving equations, identifying real solutions is often the primary goal, as they have direct applications in many real-world scenarios. However, it's crucial to remember that quartic equations can also have complex solutions, which we will explore in the next step. The process of finding real solutions involves understanding the properties of roots and using tools like calculators to obtain numerical approximations when necessary. By carefully considering both positive and negative roots, we ensure that we've identified all possible real solutions to the equation.

Step 4: Determining the Complex Solutions

While finding the real solutions is a significant part of solving the equation X⁴ - 24 = 0, a complete solution requires us to also consider complex solutions. Complex solutions are numbers that involve the imaginary unit i, where i is defined as the square root of -1. Quartic equations, being fourth-degree polynomials, have four roots in total, according to the fundamental theorem of algebra. We've already found two real roots, so we know there must be two complex roots as well. To find the complex solutions, we need to delve into the complex number system. Recall that we have X⁴ = 24. We can rewrite 24 in polar form, which is a way of representing complex numbers using magnitude and angle. However, a more direct approach for this equation involves recognizing that X⁴ = 24 can also be expressed in terms of complex roots. We know that X = ± ⁴√24 gives us the real roots. To find the complex roots, we consider that complex roots often come in conjugate pairs for polynomials with real coefficients. Let's express the complex roots in the form a + bi and a - bi, where a and b are real numbers. We can rewrite the equation X⁴ = 24 as X⁴ - 24 = 0. Factoring this as a difference of squares (though involving imaginary numbers), we can express it as (X² - ⁴√24)(X² + ⁴√24) = 0. This gives us two quadratic equations to solve: X² = ⁴√24 and X² = -⁴√24. The first equation gives us the real roots we've already found. The second equation, X² = -⁴√24, leads us to the complex roots. Taking the square root of both sides, we get X = ± √(-⁴√24). We can rewrite this as X = ± i√(⁴√24), where i is the imaginary unit. Thus, the complex solutions are X = i√(⁴√24) and X = -i√(⁴√24). These complex solutions are purely imaginary, meaning they have no real part. They are equally important as the real solutions in fully solving the quartic equation. By understanding the nature of complex numbers and their role in polynomial equations, we can confidently identify and express these complex solutions.

Step 5: Verifying the Solutions

The final step in solving the equation X⁴ - 24 = 0 is to verify the solutions we've found. This crucial step ensures that our solutions are accurate and satisfy the original equation. We have identified four solutions: two real solutions (X = ⁴√24 and X = -⁴√24) and two complex solutions (X = i√(⁴√24) and X = -i√(⁴√24)). To verify these solutions, we substitute each one back into the original equation, X⁴ - 24 = 0, and check if the equation holds true. Let's start with the real solutions. For X = ⁴√24, we have (⁴√24)⁴ - 24 = 24 - 24 = 0, which confirms that this is a valid solution. Similarly, for X = -⁴√24, we have (-⁴√24)⁴ - 24 = 24 - 24 = 0, also confirming its validity. Now, let's verify the complex solutions. For X = i√(⁴√24), we have (i√(⁴√24))⁴ - 24. This simplifies to i⁴(√(⁴√24))⁴ - 24. Since i² = -1, then i⁴ = (i²)² = (-1)² = 1. Also, (√(⁴√24))⁴ = (⁴√24)², so we have (⁴√24)² - 24. However, this should be 1 * (-24) - 24 = -24 -24 which is not zero. There seems to be an error in the previous calculation of complex roots. Going back to X² = -⁴√24, taking square root on both sides give X = ±√(-⁴√24) = ± i√(⁴√24). Let a = ⁴√24, then complex solutions are X = ± i√a. Thus, X⁴ = (± i√a)⁴ = (i√a)⁴ = i⁴ (√a)⁴ = 1 * a² = (⁴√24)² = √24. Substituting into the original equation: √24 - 24 = 0 which is incorrect. We need to revise the complex root calculations. Since X⁴ = 24, we can write 24 = 24(cos(0) + i sin(0)). In polar form, complex roots are given by X_k = ⁴√24 [cos((2πk + 0)/4) + i sin((2πk + 0)/4)] for k = 0, 1, 2, 3. For k = 0, X_0 = ⁴√24 (cos(0) + i sin(0)) = ⁴√24, the positive real root. For k = 1, X_1 = ⁴√24 (cos(π/2) + i sin(π/2)) = i ⁴√24. For k = 2, X_2 = ⁴√24 (cos(π) + i sin(π)) = -⁴√24, the negative real root. For k = 3, X_3 = ⁴√24 (cos(3π/2) + i sin(3π/2)) = -i ⁴√24. Let's test X = i ⁴√24: (i ⁴√24)⁴ - 24 = i⁴ (⁴√24)⁴ - 24 = 1 * 24 - 24 = 0, which is correct. Similarly, for X = -i ⁴√24: (-i ⁴√24)⁴ - 24 = (-i)⁴ (⁴√24)⁴ - 24 = 1 * 24 - 24 = 0, which is also correct. This verification process highlights the importance of checking our work to ensure accuracy. By substituting each solution back into the original equation, we gain confidence in our results and demonstrate a thorough understanding of the solving process. This step is not merely a formality but a critical component of mathematical problem-solving.

Conclusion: Summarizing the Solutions

In this comprehensive guide, we have meticulously walked through the process of solving the quartic equation X⁴ - 24 = 0. We began by isolating the variable term, then took the fourth root of both sides, and subsequently identified both real and complex solutions. The final step involved verifying these solutions to ensure their accuracy. To summarize, the solutions to the equation X⁴ - 24 = 0 are as follows: The real solutions are X = ⁴√24 and X = -⁴√24, which are approximately 2.213 and -2.213, respectively. The complex solutions are X = i⁴√24 and X = -i⁴√24. These four solutions represent the complete set of roots for the given quartic equation. Understanding how to solve quartic equations like this one is a valuable skill in algebra. It requires a combination of algebraic manipulation, knowledge of roots and radicals, and an understanding of complex numbers. The step-by-step approach we've employed here can be applied to other similar equations, providing a systematic way to find solutions. Moreover, the process of verifying solutions is a crucial practice in mathematics, ensuring that our answers are correct and that we have a solid grasp of the concepts involved. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems. This exploration into solving quartic equations demonstrates the beauty and complexity of mathematics, where each step builds upon the previous one to reveal the final answer. The journey from the initial equation to the verified solutions is a testament to the power of algebraic methods and the importance of careful, methodical problem-solving.