If ` 2 + x sqrt(3) = (1)/( 2 + sqrt( 3))` then the simplest value of x is

Options

A. `-1`
B. `1`
C. `-2`
D. `2`

Answer

To solve the equation \( 2 + x \sqrt{3} = \frac{1}{2 + \sqrt{3}} \), we will follow these steps:

### Step 1: Rationalize the Right-Hand Side
We start with the equation:
\[ 2 + x \sqrt{3} = \frac{1}{2 + \sqrt{3}} \]
To simplify the right-hand side, we will rationalize it by multiplying 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 2: Simplify the Denominator
Now, we simplify the denominator using the difference of squares:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus, the right-hand side simplifies to:
\[ \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3} \]

### Step 3: Set the Equation
Now we can rewrite the equation:
\[ 2 + x \sqrt{3} = 2 - \sqrt{3} \]

### Step 4: Isolate the \( x \sqrt{3} \) Term
Subtract 2 from both sides:
\[ x \sqrt{3} = -\sqrt{3} \]

### Step 5: Solve for \( x \)
Now, divide both sides by \( \sqrt{3} \):
\[ x = -1 \]

### Conclusion
The simplest value of \( x \) is:
\[ \boxed{-1} \]