Solve For K: Rewriting A Quadratic Expression

Let's dive into a classic algebra problem! We're given the expression $\frac{1}{3}x^2 - 2$ and told it can be rewritten in the form $\frac{1}{3}(x-k)(x+k)$, where $k$ is a positive constant. Our mission? To find the value of $k$. This type of problem is all about understanding how to manipulate and recognize different forms of quadratic expressions. We'll break it down step-by-step, making sure everything is crystal clear. This is a great exercise in applying the difference of squares pattern and understanding how constants affect the shape and position of a parabola. Mastering this skill will not only help you ace your math exams but also give you a solid foundation for more advanced mathematical concepts. By working through this problem, you'll gain a better grasp of algebraic manipulation and how to relate different forms of the same expression. We will use techniques that are widely applicable in mathematics, so pay close attention.

Understanding the Problem

First, let's make sure we've got a firm grasp of the problem. We're starting with $\frac{1}{3}x^2 - 2$ and we know it's equivalent to $\frac{1}{3}(x-k)(x+k)$. The key here is that $k$ is a positive number. Our goal is to figure out exactly what value of $k$ makes these two expressions identical. The expression $\frac{1}{3}(x-k)(x+k)$ looks suspiciously like the difference of squares formula in disguise. Remember that the difference of squares formula states that $a^2 - b^2 = (a-b)(a+b)$. This is a crucial concept in algebra, and it's something you'll encounter repeatedly. Recognizing this pattern is the first step toward solving our problem. So, let's explore how we can connect our initial expression to this formula. We're looking to rewrite the original expression in a way that allows us to easily identify the value of $k$. The problem presents us with two equivalent forms of the same quadratic expression. Our task is to determine the unknown constant $k$. This involves algebraic manipulation and the application of key mathematical concepts such as the difference of squares. This isn't just about finding the answer; it's about understanding the why behind the solution.

Applying the Difference of Squares

Let's work with the expression $\frac{1}{3}(x-k)(x+k)$. If we expand this, we get $\frac{1}{3}(x^2 - k^2)$. Now, we want to make this look like our original expression, $\frac{1}{3}x^2 - 2$. Notice that both expressions have a $\frac{1}{3}x^2$ term. That's a good sign! The next step is to make the constant terms match up. We need to find a value for $k$ such that $\frac{1}{3}k^2$ is equal to $2$. The way we'll do this is by first multiplying both sides of the equation by 3. This will help us isolate $k^2$ and get a cleaner equation to work with. Remember, the goal is to isolate $k$ so that we can clearly see its value. It's a game of algebraic manipulation, and each step should bring us closer to the solution. The difference of squares is your friend here. We want to apply it in reverse to make sure we're correctly matching the original and rewritten forms of our equation. It is also important to remember that we're looking for a positive value for $k$. This detail is critical because it narrows down our possible solutions. Keep in mind that when we take the square root, we consider both positive and negative results, but in this case, we are only going to consider the positive one.

Solving for k

Now, let's formally solve for $k$. We have $\frac{1}{3}k^2 = 2$. Multiply both sides by 3 to get $k^2 = 6$. Now, take the square root of both sides. This gives us $k = \pm \sqrt{6}$. But, remember, the problem states that $k$ is a positive constant. Therefore, $k = \sqrt{6}$. That's it! We've found our answer. Now, let's recap our steps. We started with the expanded form of $\frac{1}{3}(x-k)(x+k)$. We then noticed the similarity to the difference of squares and expanded the expression. After expanding, we set the constant terms equal to each other and solved for $k$. This involved some simple algebraic manipulation – multiplying, squaring, and taking the square root. Always double-check your work to ensure you haven't made any careless mistakes, especially when dealing with signs and square roots. This problem highlights the importance of recognizing patterns in algebra. If you can quickly identify the difference of squares or other common algebraic identities, you'll be well on your way to solving more complex problems. It also shows you how a simple change in representation can illuminate the underlying structure of a mathematical expression.

Verification of the Solution

To make sure we're right, let's plug our value of $k$ back into the rewritten expression: $\frac{1}{3}(x - \sqrt{6})(x + \sqrt{6})$. Expanding this, we get $\frac{1}{3}(x^2 - 6)$, which simplifies to $\frac{1}{3}x^2 - 2$. This matches our original expression, so we know we've got the right answer. Verification is a crucial step in any mathematical problem. This helps prevent silly errors and confirms that you have accurately found the solution. Always take the time to substitute your answer back into the original equation, or use another method to prove that the solution is correct. This is great practice for not only this type of problem, but all the mathematical problems you will encounter! In this case, verification served to confirm our work, showcasing that our application of algebraic principles resulted in the correct solution. It provides assurance that our manipulations aligned with the problem's conditions and that our understanding of the underlying concepts was accurate. Always take the time to verify your solution.

Conclusion

Therefore, the value of $k$ is $\sqrt{6}$, which corresponds to option D. This problem demonstrates a practical application of the difference of squares formula and highlights the importance of algebraic manipulation. Always pay close attention to the details of the question, such as whether $k$ needs to be a positive constant or not. This is a very common type of question you'll see on standardized tests and in your math classes. Remember to practice these types of problems regularly to improve your skills. You can also vary the problem by altering the coefficients or constants and solving for different variables. The key is to practice, practice, practice! Through consistent practice, you'll become more confident and proficient in algebra and other mathematical topics.

For further practice and understanding of quadratic equations, you can explore resources like Khan Academy's Algebra Section. This resource offers numerous examples, exercises, and videos to enhance your understanding. By reviewing these concepts and practicing frequently, you'll build a solid foundation in algebra and be well-prepared for any related problems you encounter.