If `x = 2+ sqrt(3)` the `(x + (1)/(x))` equals

Options

A. `-2sqrt(3)`
B. 2
C. 4
D. `4-2sqrt(3)`

Answer

To solve the problem where \( x = 2 + \sqrt{3} \) and we need to find \( x + \frac{1}{x} \), we can follow these steps:

### Step 1: Write down the expression
We start with the expression we want to evaluate:
\[ x + \frac{1}{x} \]

### Step 2: Substitute the value of \( x \)
Substituting \( x = 2 + \sqrt{3} \) into the expression gives us:
\[ (2 + \sqrt{3}) + \frac{1}{2 + \sqrt{3}} \]

### Step 3: Simplify \( \frac{1}{2 + \sqrt{3}} \)
To simplify \( \frac{1}{2 + \sqrt{3}} \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( 2 - \sqrt{3} \):
\[ \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} \]

### Step 4: Calculate the denominator
Now, we calculate the denominator:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus, we have:
\[ \frac{1}{2 + \sqrt{3}} = 2 - \sqrt{3} \]

### Step 5: Substitute back into the expression
Now we substitute back into our expression:
\[ x + \frac{1}{x} = (2 + \sqrt{3}) + (2 - \sqrt{3}) \]

### Step 6: Combine like terms
Combining the terms:
\[ (2 + 2) + (\sqrt{3} - \sqrt{3}) = 4 + 0 = 4 \]

### Final Answer
Thus, the value of \( x + \frac{1}{x} \) is:
\[ \boxed{4} \]