Solving For X: Unraveling The Equation X⁴ - 100x² = 0

Hey math enthusiasts! Let's dive into a classic algebra problem. We're tasked with solving for x in the equation x⁴ - 100x² = 0. This might look a bit intimidating at first, with that x raised to the fourth power, but trust me, it's totally manageable. We'll break it down step by step, making sure everyone understands how to arrive at the correct answer. This is a fantastic opportunity to flex our factoring skills and refresh our understanding of how equations work. By the end of this, you'll not only have the solution but also a deeper appreciation for the elegance of algebra. Let's get started! I'll show you the easiest way to get to the correct answer. This problem is a great way to show off your algebra skills. So let's begin and break down the equation to find x.

Understanding the Problem and Initial Steps

Alright, so the core of our problem is the equation x⁴ - 100x² = 0. Our goal is clear: find the values of x that make this equation true. What does that actually mean? It means we're looking for the numbers that, when plugged into the equation, result in zero. This type of problem is often solved using factoring techniques. Factoring is like taking apart a complex Lego structure into its simpler components. In this case, we're looking to simplify our equation into a form where we can easily identify the solutions. The initial step is to look for common factors. Notice that both terms in our equation, x⁴ and 100x², have x² in common. We can factor out x² from both terms. Let's do it step by step. First, we can rewrite the equation as x²(x² - 100) = 0. We've effectively divided both terms by x² and pulled it out front. This simplifies our equation significantly and sets us up for the next phase. This is pretty cool, right? We took a complicated-looking equation and made it a bit more friendly. From here, we can start to see where we are headed with this. The goal is always to simplify and find x.

Factoring the Equation

Now that we have our equation in the form x²(x² - 100) = 0, we are one step closer to finding the values of x. The term (x² - 100) looks very familiar. In fact, it's a difference of squares. Remember the difference of squares formula: a² - b² = (a - b)(a + b)? We can apply it here, where a is x and b is 10. So, (x² - 100) can be factored into (x - 10)(x + 10). Now, our equation becomes x²(x - 10)(x + 10) = 0. This is a beautiful thing! We've now broken down our original equation into three factors. For the equation to equal zero, at least one of these factors must be equal to zero. This is the fundamental principle we will use to find our solutions. To find the solutions, we need to set each factor equal to zero and solve for x. Let's take it one step at a time. We can solve each factor independently to get our answers. This is a fun way to approach the equation, and now you can really see where it is going.

Finding the Solutions

We have our factored equation: x²(x - 10)(x + 10) = 0. As mentioned before, to find the solutions, we set each factor equal to zero. Let's start with the first factor, x² = 0. To solve for x, we take the square root of both sides, which gives us x = 0. This is our first solution. Easy peasy, right? Next, let's consider the second factor, (x - 10) = 0. Adding 10 to both sides gives us x = 10. This is our second solution. Now, let's look at the third factor, (x + 10) = 0. Subtracting 10 from both sides, we get x = -10. This is our third solution. So, we've found three solutions: x = 0, x = 10, and x = -10. But wait, there is more! Now, let's examine the answer choices provided to see which one matches our solutions. This is important because this allows us to determine the correct answer and show that we understood the material. We will be able to easily see the correct answer once we look at the choices.

Matching the Solution to the Answer Choices

We've done the hard part: solving for x. Now, we just need to match our solutions to the answer choices. Our solutions are x = 0, x = 10, and x = -10. Let's look at the options provided:

A. 14 B. 0 C. 13 D. -3

From the provided choices, we can see that option B, which is 0, matches one of our solutions. Therefore, the correct answer is B. 0. Boom! We solved the equation and found the correct answer. This is a great feeling, and now you know how to solve this type of problem! This is a step-by-step guide, and you should be able to solve many of these types of problems now. Remember, understanding the process is more important than just getting the right answer, and now you will be able to go out and solve these types of questions with ease. It's all about breaking down the problem and using the right tools. Remember to practice these steps, and you will be able to solve the equations yourself!

Conclusion

We did it, guys! We successfully solved the equation x⁴ - 100x² = 0 and found that one of the solutions is 0. We started with a seemingly complex equation and broke it down using factoring, a difference of squares, and basic algebraic principles. This process highlighted the power of simplifying and understanding the fundamentals of algebra. The journey through this problem reinforces the importance of recognizing patterns, applying the right formulas, and, of course, practicing! Keep at it, and your algebra skills will continue to grow. Remember, the key to mastering math is consistency and a willingness to approach each problem with curiosity. Thanks for joining me on this mathematical adventure. Keep practicing, and you will become a pro! This exercise should give you a foundation for solving more complex equations. You should always remember the steps to follow when doing problems like this. And of course, keep doing math, and you will get better!