Solving (a + √b)(a - √b) And Finding The Value Of P
This comprehensive article delves into the solutions of two intriguing mathematical problems. First, we will explore the simplification of the expression (a + √b)(a - √b). Second, we will determine the value of p given the equation x = √7/5 and 5/x = p√7. These problems highlight fundamental algebraic principles and demonstrate how to manipulate expressions to arrive at concise and accurate answers. Whether you are a student looking to enhance your understanding or simply a math enthusiast, this article provides clear explanations and step-by-step solutions to these interesting questions.
Solving (a + √b)(a - √b)
The expression (a + √b)(a - √b) might seem complex at first glance, but it beautifully demonstrates a common algebraic identity. To effectively tackle this problem, the primary concept to grasp is the difference of squares. This algebraic identity is a cornerstone in simplifying expressions and solving equations. The difference of squares states that (x + y)(x - y) = x² - y². Understanding this pattern allows us to quickly simplify expressions that fit this format, saving time and reducing the likelihood of errors in more complex calculations. In this specific scenario, our expression perfectly aligns with this identity, making it straightforward to simplify. Recognizing such patterns is an essential skill in algebra, as it streamlines the problem-solving process and provides a clear path to the solution.
Applying the difference of squares identity is the key to simplifying (a + √b)(a - √b). Let's break it down: here, a corresponds to x and √b corresponds to y in the identity (x + y)(x - y) = x² - y². By directly substituting a for x and √b for y, we can rewrite the expression using the identity. This substitution transforms our original expression into a much simpler form. The beauty of using this identity lies in its ability to reduce a seemingly complex multiplication problem into a basic subtraction, significantly simplifying the mathematical process. This not only makes the problem easier to solve but also reduces the chances of making errors during manual calculation. Therefore, the difference of squares identity is not just a formula but a powerful tool in simplifying algebraic expressions efficiently.
Substituting a and √b into the difference of squares identity, we get (a + √b)(a - √b) = a² - (√b)². The next crucial step is to simplify the term (√b)². Recall that squaring a square root effectively cancels out the square root operation. This is because the square root of a number, when squared, returns the original number. In mathematical terms, (√x)² = x. Applying this principle to our expression, (√b)² simplifies to b. This simplification is a direct consequence of the fundamental properties of square roots and exponents. Understanding this relationship is vital for manipulating expressions involving radicals. It allows us to move from a more complex radical expression to a simpler, more manageable form, which is crucial for solving equations and further simplifying algebraic problems. This step underscores the importance of having a solid grasp of basic mathematical principles when tackling more complex problems.
Therefore, by simplifying (√b)² to b, the expression (a + √b)(a - √b) becomes a² - b. This final form represents the simplified version of the original expression. The entire simplification process showcases the power and efficiency of the difference of squares identity. By recognizing this pattern early on, we avoided the need for tedious multiplication and arrived at the solution swiftly. This result, a² - b, is not only the answer to the problem but also a demonstration of how algebraic identities can streamline mathematical problem-solving. The simplified expression is concise and clear, making it easier to use in further calculations or applications. This illustrates the importance of mastering algebraic identities, as they provide shortcuts and simplify complex mathematical manipulations, making them an indispensable tool in algebra.
Determining the Value of p
Now, let's shift our focus to the second part of the problem: finding the value of p given that x = √7/5 and 5/x = p√7. This problem involves algebraic manipulation and substitution to isolate and determine the unknown variable, p. To begin, it's important to understand the given equations and how they relate to each other. We have an explicit value for x, and we have an equation that includes x, p, and a square root. The key to solving this problem is to use the known value of x to substitute it into the second equation and then solve for p. This process of substitution is a fundamental technique in algebra, allowing us to simplify equations and solve for unknowns by replacing variables with their known values. Mastering this technique is essential for tackling more complex algebraic problems and is a cornerstone of mathematical problem-solving.
The first step in finding the value of p is to substitute the given value of x, which is √7/5, into the equation 5/x = p√7. Substituting x with √7/5 transforms the equation into 5/(√7/5) = p√7. This substitution is a critical step because it replaces the variable x with a specific value, allowing us to work towards isolating p. Once x is replaced, the equation now only contains p as an unknown, which means we can manipulate the equation to solve for p. The process of substitution is not just about replacing a variable; it's about simplifying the equation and making it solvable. This skill is invaluable in algebra and is frequently used in various mathematical contexts to solve for unknown variables in a system of equations or individual expressions.
Next, we need to simplify the left side of the equation, which is 5/(√7/5). Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, 5/(√7/5) can be rewritten as 5 * (5/√7). This transformation is based on a fundamental principle of arithmetic: dividing by a number is equivalent to multiplying by its inverse. Understanding this principle is crucial for simplifying complex fractions and is a common technique used in various mathematical calculations. By converting the division into multiplication, we make the expression easier to handle. This simplification step is not just a matter of calculation; it’s a strategic move that allows us to rearrange and solve the equation more efficiently, setting the stage for further simplification and ultimately finding the value of p.
Multiplying 5 by (5/√7), we get 25/√7. Now the equation looks like 25/√7 = p√7. To isolate p, we need to get rid of the √7 on the right side of the equation. We can do this by dividing both sides of the equation by √7. This algebraic manipulation is based on the principle that performing the same operation on both sides of an equation maintains the equality. Dividing by √7 is a strategic step because it will cancel out the √7 term on the right side, effectively isolating p. This technique of maintaining equality by performing identical operations on both sides is a fundamental concept in algebra. It allows us to rearrange equations and solve for unknown variables by systematically eliminating terms that are hindering the isolation of the variable we are solving for. Understanding and applying this principle is crucial for solving algebraic equations efficiently and accurately.
Dividing both sides of the equation 25/√7 = p√7 by √7 gives us (25/√7) / √7 = (p√7) / √7. Simplifying the right side, (p√7) / √7 becomes p, as the √7 terms cancel each other out. On the left side, we have (25/√7) / √7, which can be rewritten as 25/(√7 * √7). Recall that √7 * √7 = 7, so the left side simplifies to 25/7. Thus, our equation now becomes 25/7 = p. This simplification process demonstrates how dividing by a square root and then simplifying using the properties of square roots can lead to a clear and concise result. The steps taken here are crucial for isolating the variable p and highlight the importance of understanding the rules of algebraic manipulation and simplification.
Therefore, the value of p is 25/7. This solution is the culmination of a series of algebraic manipulations, including substitution, simplification of fractions, and isolating the variable. The process underscores the importance of understanding the fundamental principles of algebra and how to apply them to solve problems. From substituting the value of x to simplifying the equation and finally isolating p, each step was crucial in arriving at the correct answer. This solution not only provides the value of p but also demonstrates a methodical approach to solving algebraic problems, which can be applied to various mathematical scenarios. The result, p = 25/7, is the final answer and the solution to the problem.
In conclusion, we have successfully solved two distinct mathematical problems. First, we simplified the expression (a + √b)(a - √b) to a² - b using the difference of squares identity. Second, we determined the value of p to be 25/7, given the equations x = √7/5 and 5/x = p√7. These solutions demonstrate the power of algebraic manipulation and the importance of understanding fundamental mathematical principles. By applying these concepts, complex expressions can be simplified, and unknown variables can be found with clarity and precision. These problem-solving techniques are essential tools in mathematics and can be applied to a wide range of scenarios.
