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The given equation is:

\[
k = \frac{x + y + 2}{xy + x + y + 1}
\]
where \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\).

We need to find the value of \(k\) by substituting the given values of \(x\) and \(y\).

Let's calculate this step by step.The value of \( k \) is \( 1 \) after substituting \( x = 2 + \sqrt{3} \) and \( y = 2 - \sqrt{3} \) into the equation.

Would you like a detailed breakdown of the steps? Here are some related questions for deeper understanding:

1. How do you simplify expressions involving square roots?
2. Why does the expression simplify to 1 in this case?
3. What are conjugates, and how are they used in simplifying products like \( (2 + \sqrt{3})(2 - \sqrt{3}) \)?
4. How can you verify the result using another method?
5. What is the importance of rationalizing expressions in algebra?

**Tip**: When dealing with expressions like \( x = 2 + \sqrt{3} \) and \( y = 2 - \sqrt{3} \), conjugates often simplify square root terms effectively.

k = (x + y + 2) / (xy + x + y + 1). Si x = 2 + √3 e y = 2 - √3, entonces k es igual a:

Learn how to solve the algebraic expression k = (x + y + 2) / (xy + x + y + 1) with values x = 2 + √3 and y = 2 - √3. Explore concepts like simplifying rational expressions, square roots, and the difference of squares. Suitable for Grades 10-12.

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