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To solve this problem, we start by equating the given expression to the proposed quadratic form and solve for \( k \).

### Given Expression:
\[
\frac{1}{3} x^2 - 2
\]

### Proposed Form:
\[
\frac{1}{3}(x-k)(x+k)
\]

Expanding the proposed form:
\[
\frac{1}{3}(x^2 - k^2) = \frac{1}{3}x^2 - \frac{1}{3}k^2
\]

We are tasked with matching this expanded form with the original expression. So, we have:
\[
\frac{1}{3}x^2 - \frac{1}{3}k^2 = \frac{1}{3}x^2 - 2
\]

### Equating Coefficients:
For the quadratic terms to match perfectly, the constants on the right side of the equation must also match:
\[
-\frac{1}{3}k^2 = -2
\]

### Solve for \( k \):
To find \( k \), solve the equation:
\[
\frac{1}{3}k^2 = 2
\]
\[
k^2 = 2 \times 3 = 6
\]
\[
k = \sqrt{6}
\]

### Conclusion:
The value of \( k \) that makes the expression equivalent is \( \sqrt{6} \). Therefore, the answer is **D) \( \sqrt{6} \)**.

Would you like to explore more about factoring or rewriting quadratic expressions? Here are some related questions:

1. How do you factor a quadratic expression where the coefficient of \( x^2 \) is greater than 1?
2. What methods can be used to find the roots of a quadratic equation?
3. How do the roots relate to the constants in the vertex form of a quadratic equation?
4. How does completing the square work to convert the standard form of a quadratic equation to vertex form?
5. What is the significance of the discriminant in a quadratic equation?
6. Can you derive the quadratic formula using the method of completing the square?
7. How do changes in the value of \( k \) affect the graph of the function \( y = x^2 - k^2 \)?
8. How do you determine if a quadratic expression can be rewritten in the form \( (x-a)(x+b) \)?

**Tip:** When comparing coefficients in algebraic expressions, ensure that each term on one side of the equation perfectly matches its counterpart on the other side to maintain equality throughout the transformation process.

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

Learn how to solve for k in the quadratic expression (1/3)x^2 - 2 rewritten as (1/3)(x - k)(x + k). This algebra problem focuses on factoring using the difference of squares and is suitable for students in Grades 9-11.

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