Solving Algebraic Expressions With Radicals Finding The Value Of (x^2+xy+y^2)/(x^2-xy+y^2) And 3x^2-5xy+3y^2

Jul 12, 2025 by ADMIN 109 views

Unlocking the Secrets of Algebraic Expressions A Deep Dive into Solving for (x^2+xy+y^2)/(x^2-xy+y^2) and 3x^2-5xy+3y^2

Introduction

In the realm of mathematics, algebraic expressions often present themselves as puzzles, challenging us to unravel their complexities and discover the underlying solutions. This article delves into a specific problem involving radicals and algebraic manipulation, aiming to provide a comprehensive understanding of how to solve for the values of two intricate expressions. Our journey begins with the given values of x and y, expressed in terms of square roots, and culminates in finding the numerical values of the expressions (x^2+xy+y2)/(x^2-xy+y2) and 3x^2-5xy+3y2. This exploration not only sharpens our algebraic skills but also highlights the elegance and precision inherent in mathematical problem-solving. Join us as we dissect each step, ensuring clarity and insight into the methods employed. This article aims to serve as a valuable resource for students, educators, and anyone with a keen interest in mathematics, offering a blend of detailed explanations and practical applications.

Defining the Variables x and y

To begin, let's clearly define the variables that form the foundation of our problem. We are given that x and y are expressed as fractions involving square roots. Specifically, x equals the fraction (√5 - √3) / (√5 + √3), and y equals the fraction (√5 + √3) / (√5 - √3). These expressions, at first glance, might seem complex, but they hold a symmetrical beauty that we will exploit to simplify our calculations. The key to effectively working with such expressions lies in the process of rationalization, a technique used to eliminate radicals from the denominator. By understanding and applying rationalization, we can transform these fractions into a more manageable form, paving the way for subsequent algebraic manipulations. This initial step is crucial, as it sets the stage for the entire problem-solving process. Before we dive into the rationalization process, it's worth noting the reciprocal relationship between x and y. Recognizing that y is simply the reciprocal of x offers a valuable shortcut that will simplify our calculations later on. This symmetry is not just a mathematical curiosity; it's a powerful tool that can significantly reduce the complexity of the problem. In the following sections, we will explore the process of rationalization in detail, demonstrating how it transforms the expressions for x and y into a form that is easier to work with.

Rationalizing x and y

To effectively tackle the expressions involving x and y, the first crucial step is rationalizing their denominators. Rationalization is a mathematical technique that eliminates radical expressions from the denominator of a fraction, making it simpler to work with. For the given value of x which is (√5 - √3) / (√5 + √3), we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (√5 + √3) is (√5 - √3). This process leverages the difference of squares identity, (a + b)(a - b) = a^2 - b^2, to eliminate the square roots from the denominator. By multiplying both the numerator and denominator by (√5 - √3), we transform x into a more manageable form. The same principle applies to y, which is given as (√5 + √3) / (√5 - √3). Here, the conjugate of the denominator (√5 - √3) is (√5 + √3). Multiplying both the numerator and the denominator by this conjugate will rationalize the denominator of y. This symmetry in the process highlights a key aspect of the problem: the reciprocal relationship between x and y. After rationalizing both x and y, we will observe that the denominators become rational numbers, simplifying further calculations. This rationalization step is not just a mathematical trick; it’s a fundamental technique in algebra that allows us to manipulate expressions more easily. Once we have rationalized x and y, we can proceed to calculate the values of the given expressions, armed with simplified forms that make the algebraic manipulations less cumbersome. The process of rationalization transforms the original expressions into a form that is more amenable to further calculations.

Calculating the Values of x and y After Rationalization

Having outlined the process of rationalization, let's now delve into the actual calculations to determine the values of x and y after this transformation. Starting with x = (√5 - √3) / (√5 + √3), we multiply both the numerator and denominator by the conjugate of the denominator, which is (√5 - √3). This gives us x = [(√5 - √3) * (√5 - √3)] / [(√5 + √3) * (√5 - √3)]. Expanding the numerator, we get (√5 - √3)^2 = (√5)^2 - 2(√5)(√3) + (√3)^2 = 5 - 2√15 + 3 = 8 - 2√15. Expanding the denominator using the difference of squares, we get (√5 + √3)(√5 - √3) = (√5)^2 - (√3)^2 = 5 - 3 = 2. Therefore, x = (8 - 2√15) / 2, which simplifies to x = 4 - √15. Now, let's turn our attention to y = (√5 + √3) / (√5 - √3). We multiply both the numerator and denominator by the conjugate of the denominator, which is (√5 + √3). This gives us y = [(√5 + √3) * (√5 + √3)] / [(√5 - √3) * (√5 + √3)]. Expanding the numerator, we get (√5 + √3)^2 = (√5)^2 + 2(√5)(√3) + (√3)^2 = 5 + 2√15 + 3 = 8 + 2√15. The denominator, as before, simplifies to (√5)^2 - (√3)^2 = 5 - 3 = 2. Therefore, y = (8 + 2√15) / 2, which simplifies to y = 4 + √15. We now have the simplified values of x and y after rationalization: x = 4 - √15 and y = 4 + √15. These simplified values will make the subsequent calculations for the expressions (x^2+xy+y2)/(x^2-xy+y2) and 3x^2-5xy+3y2 much more manageable.

Part (a) Solving for (x^2 + xy + y^2) / (x^2 - xy + y^2)

With the simplified values of x and y in hand, we can now tackle the first part of the problem: finding the value of the expression (x^2 + xy + y^2) / (x^2 - xy + y^2). This expression may seem daunting at first, but by strategically breaking it down, we can simplify the calculations significantly. A key observation here is the presence of squares and the product xy. Before diving into direct substitution, it's often beneficial to explore if we can rewrite the expression in a more convenient form. Notice that both the numerator and the denominator contain terms that are closely related to the expansion of (x + y)^2 and (x - y)^2. Specifically, (x + y)^2 = x^2 + 2xy + y^2 and (x - y)^2 = x^2 - 2xy + y^2. We can rewrite the numerator as (x^2 + xy + y^2) = (x^2 + 2xy + y^2) - xy = (x + y)^2 - xy, and similarly, the denominator as (x^2 - xy + y^2) = (x^2 - 2xy + y^2) + xy = (x - y)^2 + xy. This transformation allows us to work with simpler terms: (x + y), (x - y), and xy. Now, let's calculate these intermediate values. We know x = 4 - √15 and y = 4 + √15, so (x + y) = (4 - √15) + (4 + √15) = 8, and (x - y) = (4 - √15) - (4 + √15) = -2√15. The product xy can be calculated as (4 - √15)(4 + √15) = 4^2 - (√15)^2 = 16 - 15 = 1. With these values, we can now substitute back into our rewritten expression. The numerator becomes (x + y)^2 - xy = 8^2 - 1 = 64 - 1 = 63, and the denominator becomes (x - y)^2 + xy = (-2√15)^2 + 1 = 4 * 15 + 1 = 60 + 1 = 61. Therefore, the value of the expression (x^2 + xy + y^2) / (x^2 - xy + y^2) is 63 / 61. This result demonstrates the power of algebraic manipulation in simplifying complex expressions. By rewriting the original expression in terms of (x + y), (x - y), and xy, we were able to reduce the computational burden and arrive at the solution efficiently.

Part (b) Solving for 3x^2 - 5xy + 3y^2

Having successfully navigated the complexities of part (a), let's now turn our attention to the second challenge: finding the value of the expression 3x^2 - 5xy + 3y^2. Similar to the previous part, a strategic approach to simplifying this expression will be key to efficient problem-solving. Before plunging into direct substitution, it's prudent to examine the structure of the expression and identify potential simplifications. Notice that the terms 3x^2 and 3y^2 share a common coefficient, which suggests the possibility of grouping them together. We can rewrite the expression as 3(x^2 + y^2) - 5xy. This rearrangement immediately sheds light on a possible pathway: expressing x^2 + y^2 in terms of known quantities. Recall the algebraic identity (x + y)^2 = x^2 + 2xy + y^2. Rearranging this, we get x^2 + y^2 = (x + y)^2 - 2xy. This transformation is a crucial step, as it allows us to express the sum of squares in terms of (x + y) and xy, both of which we have already calculated in part (a). Substituting this back into our expression, we get 3[(x + y)^2 - 2xy] - 5xy. Now, we can further simplify this to 3(x + y)^2 - 6xy - 5xy, which gives us 3(x + y)^2 - 11xy. At this juncture, we have successfully rewritten the original expression in terms of (x + y) and xy. We know from our previous calculations that (x + y) = 8 and xy = 1. Substituting these values, we get 3(8^2) - 11(1) = 3(64) - 11 = 192 - 11 = 181. Therefore, the value of the expression 3x^2 - 5xy + 3y^2 is 181. This result underscores the importance of algebraic manipulation in simplifying expressions and making calculations more manageable. By strategically rearranging the terms and leveraging known algebraic identities, we were able to efficiently determine the value of the given expression.

Conclusion

In this comprehensive exploration, we successfully unraveled the intricacies of two algebraic expressions involving radicals. By systematically applying the principles of rationalization and algebraic manipulation, we transformed seemingly complex problems into manageable calculations. We began by defining the variables x and y, expressed as fractions with square roots, and then employed the technique of rationalization to eliminate radicals from their denominators. This crucial step paved the way for simplified calculations in the subsequent parts of the problem. For part (a), we tackled the expression (x^2 + xy + y^2) / (x^2 - xy + y^2). By strategically rewriting the numerator and denominator in terms of (x + y) and (x - y), we were able to reduce the computational burden and arrive at the solution of 63/61. This demonstrated the power of algebraic manipulation in simplifying complex fractions. In part (b), we focused on the expression 3x^2 - 5xy + 3y^2. By rearranging terms and leveraging the algebraic identity (x + y)^2 = x^2 + 2xy + y^2, we transformed the expression into a form that was readily solvable. Substituting the previously calculated values of (x + y) and xy, we efficiently determined the value of the expression to be 181. Throughout this journey, we have highlighted the importance of a methodical approach to problem-solving in mathematics. By carefully analyzing the structure of expressions, identifying opportunities for simplification, and strategically applying algebraic techniques, we can conquer even the most challenging problems. This exploration not only enhances our algebraic skills but also cultivates a deeper appreciation for the elegance and precision of mathematics. The techniques and strategies discussed in this article serve as valuable tools for students, educators, and anyone with a passion for mathematical problem-solving. By mastering these skills, we empower ourselves to tackle a wide range of algebraic challenges with confidence and proficiency.