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We are given the equation:

\[
x + \frac{1}{x} = 6
\]

The task is to find \( x - \frac{1}{x} \) and match it with one of the provided choices.

### Step 1: Solve for \( x + \frac{1}{x} \)
We already know that:

\[
x + \frac{1}{x} = 6
\]

### Step 2: Square both sides
Let's square both sides of the equation to find a relationship that could help us with \( x - \frac{1}{x} \).

\[
\left( x + \frac{1}{x} \right)^2 = 6^2
\]

Expanding the left side:

\[
x^2 + 2 + \frac{1}{x^2} = 36
\]

Simplifying:

\[
x^2 + \frac{1}{x^2} = 36 - 2 = 34
\]

### Step 3: Use the identity for \( x - \frac{1}{x} \)
We use the identity:

\[
\left( x - \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} - 2
\]

From Step 2, we already know \( x^2 + \frac{1}{x^2} = 34 \). Substituting this into the equation:

\[
\left( x - \frac{1}{x} \right)^2 = 34 - 2 = 32
\]

### Step 4: Take the square root
Now, take the square root of both sides:

\[
x - \frac{1}{x} = \sqrt{32} = 4\sqrt{2}
\]

### Step 5: Conclusion
The value of \( x - \frac{1}{x} \) is \( 4\sqrt{2} \), which corresponds to option **D**.

---

Would you like a detailed explanation of any part of the solution?

Here are five related questions to deepen your understanding:
1. How would the problem change if \( x + \frac{1}{x} \) had a different value, like 8 instead of 6?
2. Can you derive other identities involving \( x + \frac{1}{x} \) and \( x - \frac{1}{x} \)?
3. What are the general applications of these identities in algebra?
4. How would you approach the same problem if it involved complex numbers?
5. What happens when you cube both sides of \( x + \frac{1}{x} = 6 \)?

**Tip:** In algebra, squaring and using identities is a powerful method for simplifying expressions with variables and their reciprocals!

x + 1/x = 6 olduğuna göre, x - 1/x ifadesi aşağıdakilerden hangisine eşit olabilir?

Learn how to solve for x - 1/x given the equation x + 1/x = 6. This algebraic problem involves using identities and the square root to find the solution. Suitable for high school students in Grades 9-12.

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