If `x=(4sqrt(2))/(sqrt(2)+1)`, then find the value of `(x+2)/(x-2)-(x+2sqrt(2))/(x-2sqrt(2))`.

Options

A. 2
B. `12+8sqrt(2)`
C. `12-8sqrt(2)`
D. `-2`

Explanation

To solve the problem, we start with the given expression for \( x \):

\[ x = \frac{4\sqrt{2}}{\sqrt{2} + 1} \]

We need to find the value of:

\[ \frac{x + 2}{x - 2} - \frac{x + 2\sqrt{2}}{x - 2\sqrt{2}} \]

### Step 1: Calculate \( x + 2 \)

First, we calculate \( x + 2 \):

\[ x + 2 = \frac{4\sqrt{2}}{\sqrt{2} + 1} + 2 \]

To add these fractions, we need a common denominator:

\[ x + 2 = \frac{4\sqrt{2}}{\sqrt{2} + 1} + \frac{2(\sqrt{2} + 1)}{\sqrt{2} + 1} \]

This simplifies to:

\[ x + 2 = \frac{4\sqrt{2} + 2\sqrt{2} + 2}{\sqrt{2} + 1} = \frac{6\sqrt{2} + 2}{\sqrt{2} + 1} \]

### Step 2: Calculate \( x - 2 \)

Next, we calculate \( x - 2 \):

\[ x - 2 = \frac{4\sqrt{2}}{\sqrt{2} + 1} - 2 \]

Again, we need a common denominator:

\[ x - 2 = \frac{4\sqrt{2}}{\sqrt{2} + 1} - \frac{2(\sqrt{2} + 1)}{\sqrt{2} + 1} \]

This simplifies to:

\[ x - 2 = \frac{4\sqrt{2} - 2\sqrt{2} - 2}{\sqrt{2} + 1} = \frac{2\sqrt{2} - 2}{\sqrt{2} + 1} \]

### Step 3: Calculate \( x + 2\sqrt{2} \)

Now, we calculate \( x + 2\sqrt{2} \):

\[ x + 2\sqrt{2} = \frac{4\sqrt{2}}{\sqrt{2} + 1} + 2\sqrt{2} \]

Using a common denominator:

\[ x + 2\sqrt{2} = \frac{4\sqrt{2}}{\sqrt{2} + 1} + \frac{2\sqrt{2}(\sqrt{2} + 1)}{\sqrt{2} + 1} \]

This simplifies to:

\[ x + 2\sqrt{2} = \frac{4\sqrt{2} + 2\cdot2 + 2\sqrt{2}}{\sqrt{2} + 1} = \frac{6\sqrt{2} + 4}{\sqrt{2} + 1} \]

### Step 4: Calculate \( x - 2\sqrt{2} \)

Next, we calculate \( x - 2\sqrt{2} \):

\[ x - 2\sqrt{2} = \frac{4\sqrt{2}}{\sqrt{2} + 1} - 2\sqrt{2} \]

Using a common denominator:

\[ x - 2\sqrt{2} = \frac{4\sqrt{2}}{\sqrt{2} + 1} - \frac{2\sqrt{2}(\sqrt{2} + 1)}{\sqrt{2} + 1} \]

This simplifies to:

\[ x - 2\sqrt{2} = \frac{4\sqrt{2} - 2\cdot2 - 2\sqrt{2}}{\sqrt{2} + 1} = \frac{2\sqrt{2} - 4}{\sqrt{2} + 1} \]

### Step 5: Substitute into the original expression

Now we substitute back into the expression we need to evaluate:

\[ \frac{x + 2}{x - 2} - \frac{x + 2\sqrt{2}}{x - 2\sqrt{2}} = \frac{\frac{6\sqrt{2} + 2}{\sqrt{2} + 1}}{\frac{2\sqrt{2} - 2}{\sqrt{2} + 1}} - \frac{\frac{6\sqrt{2} + 4}{\sqrt{2} + 1}}{\frac{2\sqrt{2} - 4}{\sqrt{2} + 1}} \]

This simplifies to:

\[ \frac{6\sqrt{2} + 2}{2\sqrt{2} - 2} - \frac{6\sqrt{2} + 4}{2\sqrt{2} - 4} \]

### Step 6: Simplify each fraction

We can simplify each fraction separately:

1. For the first term:
\[ \frac{6\sqrt{2} + 2}{2(\sqrt{2} - 1)} = \frac{3\sqrt{2} + 1}{\sqrt{2} - 1} \]

2. For the second term:
\[ \frac{6\sqrt{2} + 4}{2(\sqrt{2} - 2)} = \frac{3\sqrt{2} + 2}{\sqrt{2} - 2} \]

### Step 7: Combine the fractions

Now we have:

\[ \frac{3\sqrt{2} + 1}{\sqrt{2} - 1} - \frac{3\sqrt{2} + 2}{\sqrt{2} - 2} \]

We can find a common denominator and combine these fractions.

### Final Answer

After performing the calculations and simplifications, we arrive at the final answer:

\[ 12 + 8\sqrt{2} \]