If ` x = sqrt(7 - 4 sqrt(3))` , then `x + (1)/(x)` is equal to :

Options

A. 2
B. `3 sqrt(7)`
C. 4
D. `4 sqrt(7)`

Answer

To solve the equation \( x = \sqrt{7 - 4\sqrt{3}} \) and find the value of \( x + \frac{1}{x} \), we can follow these steps:

### Step 1: Simplify \( x \)
We start with:
\[ x = \sqrt{7 - 4\sqrt{3}} \]
To simplify this expression, we can express \( 7 - 4\sqrt{3} \) in a different form. We look for two numbers \( a \) and \( b \) such that:
\[ x = \sqrt{(a - b)^2} = a - b \]
We can assume \( a^2 + b^2 = 7 \) and \( 2ab = 4\sqrt{3} \). From \( 2ab = 4\sqrt{3} \), we get:
\[ ab = 2\sqrt{3} \]

### Step 2: Solve for \( a \) and \( b \)
We can express \( a \) and \( b \) in terms of \( a + b \) and \( ab \). Let:
\[ s = a + b \quad \text{and} \quad p = ab \]
From the equations:
\[ s^2 = a^2 + b^2 + 2ab = 7 + 4\sqrt{3} \]
We can find \( a \) and \( b \) by solving:
1. \( a + b = s \)
2. \( ab = 2\sqrt{3} \)

### Step 3: Calculate \( x + \frac{1}{x} \)
Using the identity:
\[ x + \frac{1}{x} = \frac{x^2 + 1}{x} \]
We need to find \( x^2 \):
\[ x^2 = 7 - 4\sqrt{3} \]
Thus:
\[ x + \frac{1}{x} = \frac{(7 - 4\sqrt{3}) + 1}{\sqrt{7 - 4\sqrt{3}}} \]
This simplifies to:
\[ x + \frac{1}{x} = \frac{8 - 4\sqrt{3}}{\sqrt{7 - 4\sqrt{3}}} \]

### Step 4: Rationalize the denominator
To simplify further, we multiply by the conjugate:
\[ \frac{(8 - 4\sqrt{3})(\sqrt{7 + 4\sqrt{3}})}{7 - 4\sqrt{3} + 4\sqrt{3}} = \frac{(8 - 4\sqrt{3})(\sqrt{7 + 4\sqrt{3}})}{1} \]

### Step 5: Final calculation
Now we can calculate \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = \sqrt{(x + \frac{1}{x})^2} = \sqrt{16} = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]