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We are given that \( x = 2 + \sqrt{5} \), and the problem asks us to find the value of \( x + \frac{1}{x} \).

### Step 1: Calculate \( \frac{1}{x} \)
We have \( x = 2 + \sqrt{5} \). To find \( \frac{1}{x} \), we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of \( x \).

\[
\frac{1}{x} = \frac{1}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{2 - \sqrt{5}}{(2 + \sqrt{5})(2 - \sqrt{5})}
\]

The denominator simplifies as:
\[
(2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1
\]

Thus,
\[
\frac{1}{x} = \frac{2 - \sqrt{5}}{-1} = \sqrt{5} - 2
\]

### Step 2: Add \( x \) and \( \frac{1}{x} \)

Now, add \( x = 2 + \sqrt{5} \) and \( \frac{1}{x} = \sqrt{5} - 2 \):
\[
x + \frac{1}{x} = (2 + \sqrt{5}) + (\sqrt{5} - 2)
\]
Simplifying:
\[
x + \frac{1}{x} = 2 - 2 + \sqrt{5} + \sqrt{5} = 2\sqrt{5}
\]

Thus, the value of \( x + \frac{1}{x} \) is \( 2\sqrt{5} \).

### Final Answer:
The correct option is \( 2\sqrt{5} \).

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Would you like a detailed explanation of any step? Here are some related questions for further practice:

1. What is the general method for rationalizing a denominator?
2. How can you find the sum of a number and its reciprocal?
3. What are the properties of conjugates in algebraic expressions?
4. Can you apply this method to a different quadratic irrational number?
5. How does simplifying square roots help in algebraic operations?

**Tip:** Rationalizing the denominator often simplifies complex expressions involving roots, making them easier to manipulate.

If x = 2 + √5, the value of x + 1/x is?

Find the value of x + 1/x where x = 2 + √5. Learn how to rationalize the denominator and simplify algebraic expressions. Suitable for students in Grades 9-11.

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