Finding The Positive Solution Of X² - 36 = 5x A Step-by-Step Guide
Introduction: Navigating the Realm of Quadratic Equations
In the captivating world of mathematics, quadratic equations stand as fundamental expressions that weave their way through various scientific and engineering disciplines. These equations, characterized by their highest power of two, often present us with intriguing challenges and solutions that hold significant practical implications. One such equation that piques our interest is x² - 36 = 5x. This equation, at first glance, may appear as a simple algebraic expression, but it holds within it a rich mathematical tapestry waiting to be unraveled. Our exploration today will delve into the depths of this equation, aiming to uncover its positive solution, a value of x that not only satisfies the equation but also adheres to the condition of being greater than zero. We will embark on a journey that involves rearranging the equation into its standard form, employing the powerful quadratic formula, and meticulously analyzing the roots to pinpoint the one that aligns with our quest for a positive solution. Along the way, we will encounter the elegance and precision that mathematics offers, revealing how these tools can be applied to solve real-world problems and illuminate the underlying principles that govern our universe.
Before we dive into the specifics of solving x² - 36 = 5x, let's take a moment to appreciate the broader context of quadratic equations. These equations, represented in the general form ax² + bx + c = 0, where a, b, and c are constants, are not merely abstract mathematical constructs. They are powerful tools that allow us to model and understand a wide array of phenomena. From the graceful arc of a projectile's trajectory to the intricate patterns of electrical circuits, quadratic equations provide a framework for describing and predicting behavior. The solutions to these equations, often referred to as roots, represent the points where the quadratic function intersects the x-axis, providing critical insights into the system being modeled. Understanding how to solve quadratic equations, therefore, is not just an academic exercise; it's a gateway to unlocking the secrets of the world around us.
Transforming the Equation: Setting the Stage for Solution
The first step in our quest to solve x² - 36 = 5x is to transform it into its standard quadratic form. This form, ax² + bx + c = 0, provides a structured framework that allows us to apply well-established solution techniques. To achieve this transformation, we need to manipulate the equation by moving all terms to one side, leaving zero on the other. In our case, this involves subtracting 5x from both sides of the equation. This seemingly simple step is crucial because it aligns our equation with the standard form, making it amenable to methods like factoring, completing the square, or the quadratic formula. The act of rearranging terms in an equation is a fundamental skill in algebra, highlighting the importance of maintaining balance and equivalence throughout the process. Each operation we perform must be applied equally to both sides, ensuring that the integrity of the equation is preserved. By carefully executing this transformation, we lay the foundation for a systematic and accurate solution.
Once we subtract 5x from both sides of x² - 36 = 5x, we arrive at the equation x² - 5x - 36 = 0. This is the standard form of our quadratic equation, where a = 1, b = -5, and c = -36. Now that we have the equation in this form, we can clearly identify the coefficients, which are essential for applying various solution methods. The coefficient 'a' represents the quadratic term, 'b' the linear term, and 'c' the constant term. These coefficients play a critical role in determining the shape and position of the parabola represented by the quadratic equation. The standard form not only simplifies the application of solution techniques but also provides a visual representation of the equation's characteristics. The parabola's orientation (opening upwards or downwards) is determined by the sign of 'a', and the location of the vertex and axis of symmetry can be derived from the coefficients as well. By transforming the equation into standard form, we gain a deeper understanding of its properties and prepare ourselves for the next stage of the solution process.
The Quadratic Formula: A Universal Solution
With our equation now in the standard form x² - 5x - 36 = 0, we turn to the quadratic formula, a powerful tool that provides a universal solution for any quadratic equation. This formula, derived from the method of completing the square, states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by: x = (-b ± √(b² - 4ac)) / 2a. The quadratic formula is a cornerstone of algebra, offering a reliable method for finding the roots of any quadratic equation, regardless of its complexity. Its beauty lies in its generality; it works even when factoring or other methods prove difficult or impossible. The formula encapsulates the relationship between the coefficients of the equation and its solutions, providing a direct path to finding the values of x that satisfy the equation. Memorizing and understanding the quadratic formula is an essential skill for anyone venturing into the realm of mathematics, as it serves as a fundamental building block for solving more advanced problems. The formula's structure also reveals important information about the nature of the solutions, as the discriminant (b² - 4ac) within the square root determines whether the roots are real, imaginary, or repeated.
To apply the quadratic formula to our equation x² - 5x - 36 = 0, we substitute the values of the coefficients a, b, and c into the formula. Recall that a = 1, b = -5, and c = -36. Plugging these values into the formula, we get: x = (-(-5) ± √((-5)² - 4 * 1 * -36)) / (2 * 1). This substitution is a critical step, requiring careful attention to signs and order of operations. The negative signs in the formula and the coefficients can be a source of errors if not handled meticulously. Once the values are substituted, the next step is to simplify the expression. We begin by simplifying the terms inside the square root, calculating (-5)² as 25 and 4 * 1 * -36 as -144. The expression then becomes: x = (5 ± √(25 + 144)) / 2. The simplification process highlights the importance of following the order of operations (PEMDAS/BODMAS), ensuring that we perform calculations in the correct sequence. By carefully substituting the values and simplifying the expression, we pave the way for finding the roots of the equation.
Calculating the Roots: Unveiling the Solutions
Continuing our journey, we now focus on simplifying the expression obtained after substituting the coefficients into the quadratic formula. We have x = (5 ± √(25 + 144)) / 2. The next step is to add 25 and 144 inside the square root, resulting in 169. Thus, the equation becomes x = (5 ± √169) / 2. The square root of 169 is 13, a perfect square, which simplifies our calculation significantly. We now have x = (5 ± 13) / 2. This expression represents two possible solutions, one obtained by adding 13 to 5 and the other by subtracting 13 from 5. These two solutions are a direct consequence of the ± sign in the quadratic formula, which arises from the nature of square roots, where both a positive and a negative value can yield the same square. Calculating these two solutions is the crux of solving the quadratic equation, as it reveals the values of x that satisfy the original equation.
To find the two solutions, we first consider the case where we add 13 to 5: x = (5 + 13) / 2 = 18 / 2 = 9. This gives us one solution, x = 9. Next, we consider the case where we subtract 13 from 5: x = (5 - 13) / 2 = -8 / 2 = -4. This gives us another solution, x = -4. We now have two roots for the equation x² - 5x - 36 = 0: x = 9 and x = -4. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis. However, our quest is not yet complete. We are specifically looking for the positive solution, a value of x that is greater than zero. The presence of two roots highlights the nature of quadratic equations, which can have two distinct real roots, one repeated real root, or two complex roots, depending on the discriminant. The process of finding these roots demonstrates the power of the quadratic formula and the importance of carefully considering all possible solutions.
Identifying the Positive Solution: The Final Step
Having calculated the two roots of the equation x² - 5x - 36 = 0, which are x = 9 and x = -4, we now focus on identifying the positive solution. Our initial problem statement specifically asked for the positive solution, adding a constraint that narrows down our answer. Among the two roots we found, 9 is a positive number, while -4 is a negative number. Therefore, the positive solution is x = 9. This final step underscores the importance of carefully rereading the problem statement and ensuring that the answer provided meets all the specified conditions. It's not enough to simply find the roots of the equation; we must also interpret them in the context of the problem and select the one that satisfies the given criteria. This process of analysis and selection is a crucial part of mathematical problem-solving, fostering critical thinking and attention to detail.
The root x = -4, while a valid solution to the equation, does not satisfy the condition of being positive. This highlights the distinction between a solution to an equation and a solution to a problem. The equation x² - 5x - 36 = 0 has two solutions, but the problem we were asked to solve had an additional constraint. This is a common theme in mathematics, where real-world problems often have constraints that limit the acceptable solutions. For example, in a physical problem, a negative length or time might not be meaningful, even if it is a solution to the underlying equation. Therefore, it's essential to not only master the techniques for solving equations but also to develop the ability to interpret the solutions in the context of the problem. By carefully considering the problem statement and the nature of the solutions, we can arrive at the correct and meaningful answer. In this case, the positive solution x = 9 is the answer we sought.
Conclusion: The Power of Mathematical Tools
In this exploration, we successfully navigated the realm of quadratic equations to uncover the positive solution of x² - 36 = 5x. Our journey began with transforming the equation into its standard form, a crucial step that paved the way for applying the quadratic formula. We then delved into the intricacies of the quadratic formula, a universal tool that empowers us to solve any quadratic equation, regardless of its complexity. By carefully substituting the coefficients and simplifying the expression, we calculated the two roots of the equation: x = 9 and x = -4. Finally, we focused on identifying the positive solution, recognizing that x = 9 is the answer that satisfies the given condition. This entire process highlights the power of mathematical tools and the importance of a systematic approach to problem-solving. Each step, from rearranging the equation to interpreting the solutions, demonstrates the elegance and precision that mathematics offers.
The skills and techniques we employed in this exploration extend far beyond the specific equation we solved. Quadratic equations are ubiquitous in mathematics and its applications, appearing in diverse fields such as physics, engineering, economics, and computer science. The ability to solve these equations is a fundamental skill that opens doors to understanding and modeling a wide range of phenomena. Moreover, the process of problem-solving itself, which involves transforming a problem into a mathematical form, applying appropriate techniques, and interpreting the results, is a valuable skill that transcends the boundaries of mathematics. By mastering these skills, we not only become proficient in mathematics but also develop critical thinking and analytical abilities that are essential for success in any field. The journey to find the positive solution of x² - 36 = 5x is a microcosm of the broader mathematical journey, one that equips us with the tools and mindset to tackle challenges and unlock the secrets of the world around us.
