If `x = ( sqrt2 - 1)/( sqrt2 + 1)`, then `( x + (1)/(x) )` is

Options

A. `6`
B. `5`
C. `3`
D. `2 sqrt2`

Answer

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

### Step 1: Simplify \( x \)
We start with:
\[ x = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \]
To simplify this expression, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{2} - 1 \).

### Step 2: Rationalize the denominator
Multiply the numerator and denominator by \( \sqrt{2} - 1 \):
\[ x = \frac{(\sqrt{2} - 1)(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} \]
The denominator simplifies to:
\[ (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \]
Thus, we have:
\[ x = (\sqrt{2} - 1)^2 \]

### Step 3: Expand \( x \)
Now, we expand \( (\sqrt{2} - 1)^2 \):
\[ x = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]

### Step 4: Find \( \frac{1}{x} \)
Next, we need to find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{3 - 2\sqrt{2}} \]
To rationalize this, we multiply the numerator and denominator by the conjugate \( 3 + 2\sqrt{2} \):
\[ \frac{1}{x} = \frac{3 + 2\sqrt{2}}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})} \]
The denominator simplifies to:
\[ 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]
Thus, we have:
\[ \frac{1}{x} = 3 + 2\sqrt{2} \]

### Step 5: Calculate \( x + \frac{1}{x} \)
Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (3 - 2\sqrt{2}) + (3 + 2\sqrt{2}) = 3 - 2\sqrt{2} + 3 + 2\sqrt{2} \]
The \( -2\sqrt{2} \) and \( +2\sqrt{2} \) cancel each other out:
\[ x + \frac{1}{x} = 3 + 3 = 6 \]

### Final Answer
Thus, the value of \( x + \frac{1}{x} \) is:
\[ \boxed{6} \]