Solving The Equation 1 + 1/x = 72/x^2 A Step-by-Step Guide
Solving equations is a fundamental skill in mathematics, and mastering different techniques is crucial for success in algebra and beyond. In this article, we will delve into the process of solving the equation 1 + 1/x = 72/x^2. This particular equation involves fractions and a quadratic form, making it a great example to illustrate various algebraic strategies. We will break down each step in detail, ensuring a clear understanding of the solution process. We will explore the initial setup, the transformation into a standard quadratic equation, and finally, the methods for finding the roots. Understanding these steps is not just about solving this one equation; it's about building a solid foundation for tackling more complex problems in the future. So, let's embark on this mathematical journey together, and by the end, you'll be well-equipped to solve similar equations with confidence.
Understanding the Problem
Before diving into the solution, it's essential to understand the nature of the equation we're dealing with. The equation 1 + 1/x = 72/x^2 is a rational equation because it involves fractions with variables in the denominator. These types of equations often require careful manipulation to avoid division by zero and to transform them into a more manageable form. A key strategy when dealing with rational equations is to eliminate the fractions. We achieve this by multiplying both sides of the equation by the least common denominator (LCD). In this case, the LCD is x^2, since it's the highest power of x appearing in the denominators. Multiplying by the LCD allows us to clear the fractions and obtain a polynomial equation, which is generally easier to solve. This step is crucial because it converts the equation into a form we are more familiar with, such as a quadratic equation. This transformation is not just a trick; it's a powerful technique that simplifies the problem significantly. By understanding this initial setup, we set the stage for a smooth and accurate solution process. The goal here is not just to find the answer but to comprehend the underlying mathematical principles that make the solution possible. So, let's move on to the next step where we'll apply this strategy and transform the equation.
Clearing the Fractions
To effectively solve the equation 1 + 1/x = 72/x^2, our initial move is to eliminate the fractions. As we discussed, this involves multiplying both sides of the equation by the least common denominator (LCD), which in this case is x^2. This step is crucial because it transforms the rational equation into a more manageable polynomial equation. When we multiply both sides by x^2, we're essentially clearing the denominators, making the equation easier to work with. Let's break down the process step by step: Multiplying the left side (1 + 1/x) by x^2 gives us x^2 * 1 + x^2 * (1/x), which simplifies to x^2 + x. On the right side, multiplying 72/x^2 by x^2 results in 72. Therefore, the equation transforms to x^2 + x = 72. Now, we have a quadratic equation in standard form. This transformation is a significant step forward because quadratic equations have well-established methods for finding solutions, such as factoring, completing the square, or using the quadratic formula. The key takeaway here is the power of algebraic manipulation in simplifying complex equations. By eliminating the fractions, we've converted the original problem into a more familiar and solvable form. This skill is invaluable in mathematics, allowing us to tackle a wide range of problems with confidence. With our equation now in the form x^2 + x = 72, we're ready to proceed to the next step: rearranging the equation into the standard quadratic form.
Rearranging into Standard Quadratic Form
Having cleared the fractions in the equation 1 + 1/x = 72/x^2, we arrived at the equation x^2 + x = 72. To solve this equation, the next critical step is to rearrange it into the standard quadratic form. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. This form is essential because it allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. To get our equation into this form, we need to move all terms to one side, leaving zero on the other side. In our case, we subtract 72 from both sides of the equation x^2 + x = 72. This gives us x^2 + x - 72 = 0. Now, the equation is in the standard quadratic form, with a = 1, b = 1, and c = -72. This rearrangement might seem like a small step, but it's a crucial one. It sets the stage for applying standard techniques to find the solutions. Once the equation is in this form, we can easily identify the coefficients and decide on the best method for solving it. In this case, factoring might be a viable option since the coefficients are relatively simple integers. However, if factoring proves difficult, we can always resort to the quadratic formula, which provides a guaranteed solution for any quadratic equation. So, with the equation now in the form x^2 + x - 72 = 0, we're well-prepared to move on to the next step: solving the quadratic equation.
Solving the Quadratic Equation
Now that we have the quadratic equation in standard form, x^2 + x - 72 = 0, we can proceed to solve it. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring appears to be the most straightforward approach. Factoring involves expressing the quadratic expression as a product of two binomials. We look for two numbers that multiply to give the constant term (-72) and add up to the coefficient of the linear term (1). These numbers are 9 and -8 because 9 * -8 = -72 and 9 + (-8) = 1. Therefore, we can factor the quadratic equation as follows: (x + 9)(x - 8) = 0. Once we have the equation in factored form, we can apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This means either x + 9 = 0 or x - 8 = 0. Solving these two linear equations gives us the solutions for x. For x + 9 = 0, subtracting 9 from both sides gives us x = -9. For x - 8 = 0, adding 8 to both sides gives us x = 8. Thus, the solutions to the quadratic equation are x = -9 and x = 8. It's always a good practice to check these solutions by substituting them back into the original equation to ensure they are valid. This helps to catch any errors made during the solving process. In this case, both solutions satisfy the original equation. Therefore, we have successfully solved the quadratic equation by factoring. However, it's important to remember that other methods, such as the quadratic formula, can be used if factoring is not feasible.
Checking the Solutions
After solving an equation, it's crucial to verify the solutions to ensure they are correct and don't lead to any inconsistencies in the original equation. This step is particularly important when dealing with rational equations, as we did in this case, because certain solutions might introduce division by zero, which is undefined. We found the solutions to the equation 1 + 1/x = 72/x^2 to be x = -9 and x = 8. To check these solutions, we substitute each value back into the original equation and see if it holds true. Let's start with x = -9: Substituting into the original equation, we get 1 + 1/(-9) = 72/(-9)^2, which simplifies to 1 - 1/9 = 72/81. Further simplification gives us 8/9 = 8/9, which is true. So, x = -9 is a valid solution. Now, let's check x = 8: Substituting into the original equation, we get 1 + 1/8 = 72/8^2, which simplifies to 1 + 1/8 = 72/64. Further simplification gives us 9/8 = 9/8, which is also true. Therefore, x = 8 is also a valid solution. Since both solutions satisfy the original equation and do not lead to any division by zero, we can confidently conclude that they are correct. Checking solutions is not just a formality; it's an essential step in the problem-solving process. It helps to avoid errors and ensures that the solutions obtained are meaningful and consistent with the original problem. In this case, our checks confirm that x = -9 and x = 8 are indeed the solutions to the given equation.
Final Answer
Having thoroughly solved and checked the equation 1 + 1/x = 72/x^2, we have arrived at the final answer. We methodically worked through each step, from clearing the fractions to factoring the quadratic equation and verifying the solutions. This process has not only given us the answer but also reinforced our understanding of algebraic techniques. The solutions we found for the equation are x = -9 and x = 8. These values satisfy the original equation without causing any inconsistencies, such as division by zero. Therefore, we can confidently state that these are the correct solutions. In the context of the multiple-choice options provided, the correct answer is A. 9, -8. It's important to note that while the process might seem lengthy for a relatively simple equation, the systematic approach we followed is applicable to a wide range of mathematical problems. By breaking down the problem into smaller, manageable steps, we can tackle even complex equations with confidence. The key takeaways from this exercise are the importance of understanding the underlying principles, the power of algebraic manipulation, and the necessity of verifying solutions. With these skills, you'll be well-equipped to solve similar equations and excel in your mathematical journey. So, let's summarize the key steps we took to arrive at this final answer.
Summary of Steps
To recap, let's summarize the key steps we took to solve the equation 1 + 1/x = 72/x^2. This will help solidify the process and provide a clear roadmap for tackling similar problems in the future. 1. Understanding the Problem: We began by recognizing the equation as a rational equation due to the presence of fractions with variables in the denominator. 2. Clearing the Fractions: We multiplied both sides of the equation by the least common denominator (LCD), which was x^2, to eliminate the fractions. This transformed the equation into x^2 + x = 72. 3. Rearranging into Standard Quadratic Form: We rearranged the equation into the standard quadratic form ax^2 + bx + c = 0 by subtracting 72 from both sides, resulting in x^2 + x - 72 = 0. 4. Solving the Quadratic Equation: We solved the quadratic equation by factoring. We identified two numbers, 9 and -8, that multiply to -72 and add up to 1. This allowed us to factor the equation as (x + 9)(x - 8) = 0. Applying the zero-product property, we found the solutions x = -9 and x = 8. 5. Checking the Solutions: We verified the solutions by substituting them back into the original equation. Both solutions satisfied the equation and did not lead to any inconsistencies. 6. Final Answer: We concluded that the solutions to the equation are x = -9 and x = 8. These steps provide a structured approach to solving rational equations and quadratic equations in general. By following this process, you can confidently tackle a wide range of mathematical problems. Remember, practice is key to mastering these techniques, so don't hesitate to apply them to various equations and scenarios. With a solid understanding of these steps, you'll be well-prepared to excel in your mathematical studies. Let's reflect on the broader implications of these techniques.
Broader Implications and Applications
The ability to solve equations like 1 + 1/x = 72/x^2 extends far beyond the classroom. These skills are fundamental to many areas of mathematics, science, engineering, and even everyday life. The techniques we've used, such as clearing fractions, rearranging equations into standard forms, and factoring, are applicable to a wide range of problems. In physics, for example, you might encounter similar equations when dealing with motion, forces, or electrical circuits. In engineering, these skills are essential for designing structures, analyzing systems, and optimizing processes. Even in fields like economics and finance, solving equations is crucial for modeling trends, making predictions, and managing resources. Moreover, the problem-solving process we've followed – understanding the problem, devising a plan, executing the plan, and reviewing the solution – is a valuable skill in any field. It encourages critical thinking, logical reasoning, and attention to detail. These are qualities that are highly valued in both academic and professional settings. So, while solving the equation 1 + 1/x = 72/x^2 might seem like a specific task, it's actually a stepping stone to developing broader problem-solving abilities. By mastering these techniques, you're not just learning how to solve equations; you're learning how to approach challenges in a systematic and effective manner. This is a skill that will serve you well throughout your life. Therefore, the time and effort invested in understanding these concepts are well worth it. Let's consider some real-world examples where these skills might come into play.
Real-World Examples
The skills we've honed in solving the equation 1 + 1/x = 72/x^2 have practical applications in numerous real-world scenarios. Understanding how to manipulate equations and solve for unknowns is a valuable asset in various fields. Let's explore a few examples to illustrate this point. 1. Engineering: Engineers often encounter equations similar to the one we solved when designing structures or analyzing systems. For instance, calculating the stress on a beam or the current in an electrical circuit might involve solving rational equations. The ability to clear fractions, rearrange terms, and apply techniques like factoring or the quadratic formula is crucial for ensuring the safety and efficiency of these designs. 2. Finance: Financial analysts use mathematical models to predict market trends, assess investment risks, and manage portfolios. These models often involve equations that need to be solved for various parameters. Understanding how to manipulate and solve equations is essential for making informed financial decisions. 3. Physics: Physics is rife with equations that describe the behavior of the physical world. Whether it's calculating the trajectory of a projectile or determining the energy levels of an atom, solving equations is a fundamental part of the discipline. The techniques we've discussed, such as clearing fractions and solving quadratic equations, are directly applicable to many physics problems. 4. Computer Science: In computer science, equations are used in algorithms, data analysis, and machine learning. For example, optimization problems often involve solving equations to find the best solution. A solid understanding of equation-solving techniques is essential for developing efficient algorithms and effective software. These are just a few examples, and the list could go on. The point is that the ability to solve equations is a versatile skill that is highly valued in many professions. By mastering these techniques, you're not just learning math; you're preparing yourself for success in a wide range of fields. Let's conclude with some final thoughts.
Final Thoughts
In conclusion, solving the equation 1 + 1/x = 72/x^2 has been more than just a mathematical exercise; it's been a journey through fundamental algebraic concepts and problem-solving strategies. We've seen how a seemingly complex equation can be broken down into manageable steps, from clearing fractions to factoring the resulting quadratic equation. We've also emphasized the importance of verifying solutions to ensure their validity. But perhaps the most significant takeaway is the realization that these skills are not confined to the realm of mathematics. They are transferable to a wide range of disciplines and real-world situations. The ability to think critically, analyze problems, and apply systematic solutions is a valuable asset in any field. So, as you continue your mathematical journey, remember that each equation you solve is not just an end in itself but also a building block for future success. The techniques you learn and the problem-solving skills you develop will serve you well in all aspects of your life. Embrace the challenges, persevere through difficulties, and celebrate your achievements. With dedication and practice, you can master any mathematical concept and unlock your full potential. And remember, the journey of a thousand miles begins with a single step. So, keep stepping forward, and never stop learning.