Simplifying (a + √b)(a - √b) And Finding Value Of P

This article delves into two intriguing mathematical problems. First, we will explore the simplification of the expression (a + √b)(a - √b), which involves the product of two binomials containing a square root. Understanding this type of expression is crucial in various areas of mathematics, including algebra and calculus. The key here is to recognize the pattern and apply the appropriate algebraic identity to simplify it efficiently. We'll break down the steps involved in the simplification process and highlight the underlying principles. Additionally, we will investigate the problem of finding the value of p given that x = √7/5 and 5/x = p√7. This problem combines concepts of radicals, fractions, and algebraic manipulation. We will carefully analyze the relationships between the variables and use algebraic techniques to isolate p and determine its value. By understanding these concepts, you will be better equipped to tackle similar mathematical challenges.

When we encounter an expression like (a + √b)(a - √b), the first thing that should come to mind is the difference of squares identity. This identity is a fundamental algebraic principle that states that (x + y)(x - y) = x² - y². Recognizing this pattern can significantly simplify the process of expanding and simplifying the expression. In our case, a corresponds to x and √b corresponds to y. Therefore, we can directly apply the identity to rewrite the expression as a² - (√b)². The square of a square root simply gives us the number inside the square root. Thus, (√b)² = b. Putting it all together, the simplified form of the expression is a² - b. This is a concise and elegant result that avoids the need for lengthy expansion and simplification steps. Understanding the difference of squares identity is not only useful for this specific problem but also for a wide range of algebraic manipulations. It's a tool that can save time and reduce the risk of errors. Furthermore, this type of simplification is essential in calculus when dealing with limits, derivatives, and integrals involving radical expressions. Mastering such fundamental algebraic principles is crucial for success in higher-level mathematics. Remember, practice is key to recognizing these patterns quickly and efficiently. By working through various examples, you'll develop an intuition for when to apply the difference of squares identity and other algebraic shortcuts. This will make you a more confident and proficient problem solver.

To determine the value of p given that x = √7/5 and 5/x = p√7, we need to use a combination of algebraic manipulation and understanding of radicals. The first step is to substitute the value of x into the second equation. This gives us 5/(√7/5) = p√7. Now, we have an equation with only one unknown, p. To solve for p, we need to isolate it on one side of the equation. Dividing by a fraction is the same as multiplying by its reciprocal. So, 5/(√7/5) can be rewritten as 5 * (5/√7), which simplifies to 25/√7. Our equation now looks like this: 25/√7 = p√7. To further simplify, we can multiply both sides of the equation by √7. This gives us 25 = p * (√7)². Since (√7)² = 7, the equation becomes 25 = 7p. Finally, to isolate p, we divide both sides of the equation by 7, resulting in p = 25/7. This value of p satisfies the given conditions. This problem demonstrates the importance of being comfortable with algebraic manipulation, including working with fractions and radicals. It also highlights the need to carefully follow the steps of solving an equation to avoid errors. By practicing problems like this, you'll strengthen your algebraic skills and gain confidence in your ability to solve more complex problems. Remember to always check your answer by substituting it back into the original equation to ensure it holds true. This is a good practice for any mathematical problem, as it can help you identify mistakes and ensure accuracy.

In this article, we tackled two distinct yet fundamental mathematical problems. We successfully simplified the expression (a + √b)(a - √b) using the difference of squares identity, arriving at the simplified form a² - b. This showcased the power of recognizing algebraic patterns and applying the appropriate identities to streamline calculations. Furthermore, we determined the value of p given the conditions x = √7/5 and 5/x = p√7. Through careful algebraic manipulation and understanding of radicals, we found that p = 25/7. Both of these problems underscore the importance of a solid foundation in algebraic principles and the ability to apply these principles in different contexts. The skills developed in solving these problems are transferable to a wide range of mathematical applications, from basic algebra to more advanced topics like calculus and differential equations. By consistently practicing and reinforcing these concepts, you will build a strong mathematical toolkit that will serve you well in your academic and professional endeavors. Remember, mathematics is a building-block subject, where each concept builds upon previous ones. Mastering the fundamentals is key to unlocking more complex and challenging problems. So, continue to explore, practice, and refine your mathematical skills, and you will be well-equipped to tackle any mathematical challenge that comes your way.