Finding The Value Of X² + Y² + Xy A Detailed Solution

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In the realm of mathematical puzzles, this problem presents a fascinating exploration of algebraic manipulation and simplification. We are given the values of x and y as expressions involving square roots, specifically x = (√3 - √2)/(√3 + √2) and y = (√3 + √2)/(√3 - √2). Our mission is to determine the value of the expression x² + y² + xy. This problem elegantly combines the concepts of rationalizing denominators, algebraic identities, and careful simplification, offering a rich opportunity to enhance our mathematical toolkit. The solution requires a strategic approach, leveraging our understanding of algebraic manipulations to navigate through the complexities of the expressions and arrive at a concise and accurate answer. This detailed exploration not only solves the specific problem but also reinforces fundamental mathematical principles applicable across a wide range of scenarios.

Rationalizing the Denominators: A Crucial First Step

The journey to finding the value of x² + y² + xy begins with simplifying the expressions for x and y. A key technique in dealing with expressions involving radicals in the denominator is rationalization. Rationalizing the denominator involves eliminating the radical from the denominator, making the expression easier to work with. For x = (√3 - √2)/(√3 + √2), we multiply both the numerator and the denominator by the conjugate of the denominator, which is (√3 - √2). This process transforms the denominator into a rational number, paving the way for further simplification. Similarly, for y = (√3 + √2)/(√3 - √2), we multiply both the numerator and the denominator by its conjugate, (√3 + √2). This initial step is critical because it simplifies the individual expressions for x and y, setting the stage for calculating x², y², and xy. Without rationalization, the subsequent calculations would be significantly more complex and prone to error. The act of rationalizing the denominator isn't just a mechanical step; it's a strategic move to make the expressions more manageable and reveal underlying relationships. This highlights the importance of recognizing and applying the appropriate algebraic techniques to solve mathematical problems efficiently.

Detailed Rationalization of x

To rationalize the denominator of x = (√3 - √2)/(√3 + √2), we multiply both the numerator and the denominator by the conjugate of the denominator, which is (√3 - √2):

x = (√3 - √2)/(√3 + √2) * (√3 - √2)/(√3 - √2)

Multiplying the numerators and denominators, we get:

x = ((√3 - √2) * (√3 - √2)) / ((√3 + √2) * (√3 - √2))

Expanding the numerator and denominator:

x = (3 - 2√6 + 2) / (3 - 2)

Simplifying:

x = (5 - 2√6) / 1

Thus, x = 5 - 2√6

Detailed Rationalization of y

Similarly, to rationalize the denominator of y = (√3 + √2)/(√3 - √2), we multiply both the numerator and the denominator by the conjugate of the denominator, which is (√3 + √2):

y = (√3 + √2)/(√3 - √2) * (√3 + √2)/(√3 + √2)

Multiplying the numerators and denominators, we get:

y = ((√3 + √2) * (√3 + √2)) / ((√3 - √2) * (√3 + √2))

Expanding the numerator and denominator:

y = (3 + 2√6 + 2) / (3 - 2)

Simplifying:

y = (5 + 2√6) / 1

Thus, y = 5 + 2√6

Calculating x², y², and xy: Building Blocks for the Solution

With the simplified expressions for x and y in hand, the next step is to calculate x², y², and xy. These individual components are the building blocks that, when combined, will lead us to the value of x² + y² + xy. Calculating x² involves squaring the expression (5 - 2√6), while y² is obtained by squaring (5 + 2√6). These calculations require careful application of the binomial expansion or the FOIL method, ensuring that each term is multiplied correctly. The product xy is found by multiplying the simplified expressions for x and y, which, interestingly, leads to a significant simplification due to the conjugate nature of the original denominators. This step highlights the interconnectedness of the problem and the strategic advantage gained by rationalizing the denominators earlier. By meticulously calculating each component, we prepare the groundwork for the final summation, demonstrating the power of breaking down a complex problem into smaller, manageable parts. This methodical approach not only ensures accuracy but also enhances our understanding of the algebraic relationships at play.

Detailed Calculation of x²

To calculate x², we square the expression x = 5 - 2√6:

x² = (5 - 2√6)²

Expanding the square:

x² = (5 - 2√6) * (5 - 2√6)

Using the FOIL method or binomial expansion:

x² = 25 - 10√6 - 10√6 + 4 * 6

Simplifying:

x² = 25 - 20√6 + 24

x² = 49 - 20√6

Detailed Calculation of y²

To calculate y², we square the expression y = 5 + 2√6:

y² = (5 + 2√6)²

Expanding the square:

y² = (5 + 2√6) * (5 + 2√6)

Using the FOIL method or binomial expansion:

y² = 25 + 10√6 + 10√6 + 4 * 6

Simplifying:

y² = 25 + 20√6 + 24

y² = 49 + 20√6

Detailed Calculation of xy

To calculate xy, we multiply the expressions x = 5 - 2√6 and y = 5 + 2√6:

xy = (5 - 2√6) * (5 + 2√6)

This is a difference of squares, so we can apply the formula (a - b)(a + b) = a² - b²:

xy = 5² - (2√6)²

Simplifying:

xy = 25 - 4 * 6

xy = 25 - 24

xy = 1

Summing the Components: Unveiling the Final Answer

Now that we have calculated x² = 49 - 20√6, y² = 49 + 20√6, and xy = 1, the final step is to add these components together to find the value of x² + y² + xy. This summation is a straightforward process, but it's crucial to ensure that we combine like terms correctly. The terms involving √6 will cancel out, leaving us with a simple integer value. This cancellation is a testament to the elegance of the problem's structure and the effectiveness of our simplification strategies. By adding the components, we complete the puzzle, revealing the numerical value of the expression x² + y² + xy. This final step underscores the importance of accuracy in each preceding calculation, as any error would propagate to the final answer. The successful completion of this step not only provides the solution but also reinforces the interconnectedness of mathematical operations and the power of careful execution.

Detailed Summation

To find the value of x² + y² + xy, we add the calculated values:

x² + y² + xy = (49 - 20√6) + (49 + 20√6) + 1

Combining like terms:

x² + y² + xy = 49 + 49 + 1 - 20√6 + 20√6

Simplifying:

x² + y² + xy = 99

Conclusion: The Elegant Solution to x² + y² + xy

In conclusion, by meticulously rationalizing denominators, calculating individual components, and carefully summing the results, we have successfully determined that the value of x² + y² + xy is 99, given x = (√3 - √2)/(√3 + √2) and y = (√3 + √2)/(√3 - √2). This journey through algebraic manipulation has not only provided a numerical answer but has also highlighted the power of strategic problem-solving techniques. The initial rationalization of denominators was a crucial step, simplifying the expressions and setting the stage for subsequent calculations. The careful calculation of x², y², and xy demonstrated the importance of accuracy in each step. Finally, the summation of these components revealed the elegant simplicity of the final answer. This problem serves as a valuable exercise in algebraic manipulation and reinforces the importance of a methodical approach in solving mathematical challenges. The elegance of the solution underscores the beauty of mathematics and the satisfaction derived from unraveling complex expressions to reveal concise and meaningful results. The process of solving this problem has not only enhanced our understanding of algebraic techniques but has also reinforced the value of precision and strategic thinking in mathematical endeavors.