The expression `1/3x^2-2` can be rewritten as `1/3(x-k)(x+k)`, where k is a positive constant . What is the value of k ?

Options

A. 2
B. 6
C. `sqrt2`
D. `sqrt6`

Explanation

To solve the problem, we need to rewrite the expression \( \frac{1}{3}x^2 - 2 \) in the form \( \frac{1}{3}(x - k)(x + k) \) and find the value of \( k \).

### Step-by-step Solution:

1. **Start with the given expression:**
\[ \frac{1}{3}x^2 - 2 \]

2. **Rewrite the constant term:**
We can express \( 2 \) as \( \frac{6}{3} \) to have a common denominator:
\[ \frac{1}{3}x^2 - \frac{6}{3} = \frac{1}{3}(x^2 - 6) \]

3. **Recognize the difference of squares:**
The expression \( x^2 - 6 \) can be rewritten using the difference of squares formula:
\[ x^2 - 6 = x^2 - (\sqrt{6})^2 \]
Therefore, we can factor it as:
\[ x^2 - 6 = (x - \sqrt{6})(x + \sqrt{6}) \]

4. **Substitute back into the expression:**
Now, substitute this back into our expression:
\[ \frac{1}{3}(x^2 - 6) = \frac{1}{3}(x - \sqrt{6})(x + \sqrt{6}) \]

5. **Identify the value of \( k \):**
From the expression \( \frac{1}{3}(x - k)(x + k) \), we can see that:
\[ k = \sqrt{6} \]

### Final Answer:
Thus, the value of \( k \) is:
\[ \boxed{\sqrt{6}} \]