Solving X² = 24 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, quadratic equations hold a significant place. These equations, characterized by the presence of a squared variable, appear frequently in various mathematical contexts and real-world applications. In this comprehensive guide, we delve into the process of solving the quadratic equation x² = 24, where x represents a real number. Our objective is not only to find the solutions but also to simplify them as much as possible, ensuring a clear and concise understanding of the solution process. We will explore the underlying concepts, step-by-step procedures, and the significance of the solutions obtained. This exploration will provide a solid foundation for tackling similar quadratic equations and appreciating their role in mathematical problem-solving.

Before diving into the solution of x² = 24, it's essential to grasp the nature of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the quadratic function intersects the x-axis when graphed. Quadratic equations have at most two real solutions, which can be found using various methods, including factoring, completing the square, and the quadratic formula. The equation x² = 24 is a special case of a quadratic equation where b = 0 and c = -24. This simplification allows for a direct approach to finding the solutions. By understanding the fundamental characteristics of quadratic equations, we can effectively apply appropriate techniques to solve them.

To solve the equation x² = 24, we employ a direct and efficient method based on the concept of square roots. The equation states that the square of x is equal to 24. To find the values of x that satisfy this condition, we need to undo the squaring operation. This is achieved by taking the square root of both sides of the equation. When taking the square root, it's crucial to remember that both positive and negative roots must be considered. This is because both the positive and negative square roots, when squared, will yield the original number. In this case, taking the square root of both sides of x² = 24 gives us √x² = ±√24. The square root of x² is simply x, so we have x = ±√24. This indicates that there are two possible solutions: the positive square root of 24 and the negative square root of 24. The next step involves simplifying the square root of 24, which will lead us to the final solutions in their simplest form. This step-by-step approach ensures a clear and accurate solution process, highlighting the importance of considering both positive and negative roots.

Having found that x = ±√24, the next step is to simplify the square root of 24. Simplification involves expressing the square root in its simplest radical form, where the number under the square root sign has no perfect square factors other than 1. To simplify √24, we first find the prime factorization of 24, which is 2 × 2 × 2 × 3 or 2³ × 3. We can then rewrite √24 as √(2² × 2 × 3). The square root of 2² is 2, so we can take the 2 out of the square root sign, leaving us with 2√(2 × 3), which simplifies to 2√6. Therefore, √24 can be simplified to 2√6. Now, substituting this simplified form back into our solutions, we get x = ±2√6. This means that the two solutions to the equation x² = 24 are x = 2√6 and x = -2√6. These solutions are in their simplest form, as √6 cannot be simplified further. The process of simplification is crucial in mathematics as it presents solutions in a clear and concise manner, making them easier to interpret and use in further calculations.

After simplifying the square root, we arrive at the two solutions to the equation x² = 24: x = 2√6 and x = -2√6. These are the two real numbers that, when squared, result in 24. The presence of two solutions is characteristic of quadratic equations, reflecting the parabolic nature of the corresponding quadratic function. The positive solution, x = 2√6, represents the point on the positive x-axis where the graph of y = x² intersects the horizontal line y = 24. Similarly, the negative solution, x = -2√6, represents the point on the negative x-axis where the same intersection occurs. These solutions are irrational numbers, as √6 is an irrational number. This means that they cannot be expressed as a simple fraction of two integers. Their decimal representations are non-repeating and non-terminating. Understanding the nature of these solutions, whether rational or irrational, is important in various mathematical contexts. The solutions x = 2√6 and x = -2√6 provide a complete and accurate answer to the problem, showcasing the power of algebraic techniques in solving equations.

To ensure the correctness of our solutions, it's always a good practice to verify them. Verification involves substituting the obtained solutions back into the original equation and checking if they satisfy the equation. Let's verify our solutions, x = 2√6 and x = -2√6, for the equation x² = 24. First, we substitute x = 2√6 into the equation: (2√6)² = 2² × (√6)² = 4 × 6 = 24. This confirms that x = 2√6 is indeed a solution. Next, we substitute x = -2√6 into the equation: (-2√6)² = (-2)² × (√6)² = 4 × 6 = 24. This also confirms that x = -2√6 is a solution. Since both solutions satisfy the original equation, we can confidently conclude that they are correct. Verification not only provides assurance about the accuracy of the solutions but also reinforces the understanding of the solution process. It's a valuable step in problem-solving, especially in mathematics, where precision is paramount.

While taking the square root is the most straightforward method for solving x² = 24, it's worth noting that other methods exist for solving quadratic equations in general. These methods include factoring, completing the square, and using the quadratic formula. Factoring involves expressing the quadratic equation as a product of two linear factors. However, this method is not directly applicable to x² = 24 as it lacks a linear term (bx). Completing the square is a technique that transforms the quadratic equation into a perfect square trinomial, making it easier to solve. While it can be applied to x² = 24, it's less efficient than taking the square root in this specific case. The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, is a universal method for solving any quadratic equation of the form ax² + bx + c = 0. Applying the quadratic formula to x² = 24 (where a = 1, b = 0, and c = -24) will also yield the solutions x = ±2√6. Understanding these alternative methods provides a broader perspective on solving quadratic equations and allows for choosing the most appropriate method based on the specific equation.

Quadratic equations are not just abstract mathematical concepts; they have numerous applications in the real world. They appear in various fields, including physics, engineering, economics, and computer science. In physics, quadratic equations are used to describe the motion of projectiles, such as the trajectory of a ball thrown in the air. The height of the projectile at any given time can be modeled using a quadratic equation. In engineering, quadratic equations are used in the design of structures, such as bridges and buildings. The shape of a parabolic arch, often used in bridge construction, is described by a quadratic equation. In economics, quadratic equations can be used to model cost, revenue, and profit functions. For example, the profit function of a business can be a quadratic equation, where the maximum profit can be found by finding the vertex of the parabola. In computer science, quadratic equations are used in various algorithms, such as those related to optimization and curve fitting. These diverse applications highlight the practical significance of quadratic equations and the importance of mastering their solutions.

In this comprehensive guide, we have successfully solved the equation x² = 24, where x is a real number. We found that the solutions are x = 2√6 and x = -2√6. We achieved this by taking the square root of both sides of the equation and simplifying the resulting square root. We also verified the solutions by substituting them back into the original equation. Furthermore, we discussed alternative methods for solving quadratic equations and explored the real-world applications of these equations. The ability to solve quadratic equations is a valuable skill in mathematics and various other fields. It requires a clear understanding of the underlying concepts, the application of appropriate techniques, and the ability to simplify solutions. By mastering these skills, one can effectively tackle a wide range of mathematical problems and appreciate the power and beauty of mathematics.