Solving X² = 100/361 A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a fun little equation to find its solutions. Our mission, should we choose to accept it (and we do!), is to solve for x in the equation x² = 100/361. Now, this might look a bit intimidating at first, but trust me, it's totally manageable. We'll break it down step-by-step, so you'll be a pro at solving these types of equations in no time. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we really understand what the equation is telling us. The equation x² = 100/361 basically asks, "What number, when multiplied by itself, equals 100/361?" Remember, squaring a number means multiplying it by itself. For example, 3 squared (3²) is 3 * 3 = 9. So, we're looking for a number that, when squared, gives us 100/361. This fraction might seem a bit scary, but don't worry, we'll handle it with ease. One crucial thing to remember when dealing with squares is that there are usually two solutions: a positive one and a negative one. This is because both a positive number and its negative counterpart, when squared, will result in a positive number. For instance, both 5² and (-5)² equal 25. Keeping this in mind will help us find all the possible solutions for x.

When we are dealing with equations like x² = 100/361, understanding the properties of squares and square roots is super important. Squaring a number is the same as raising it to the power of 2, which means multiplying the number by itself. On the flip side, the square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. But here's a tricky part: both 3 and -3, when squared, give you 9. So, technically, 9 has two square roots: 3 and -3. This is why, when we solve equations involving squares, we need to consider both the positive and negative roots. Back to our equation, 100 and 361 are perfect squares, meaning they each have an integer square root. This is a big hint that we can simplify the equation nicely. Recognizing perfect squares makes solving these types of problems way easier and faster. It’s like having a secret weapon in your math arsenal!

Thinking about fractions, it’s essential to remember how operations work with them. A fraction represents a part of a whole, and in our equation, 100/361 is the result of squaring x. The cool thing about fractions is that you can often simplify them by finding common factors in the numerator (the top number) and the denominator (the bottom number). However, in this case, 100/361 is already in its simplest form, meaning 100 and 361 don't share any common factors other than 1. When taking the square root of a fraction, we can actually take the square root of the numerator and the square root of the denominator separately. This makes the whole process much more manageable. It’s like splitting a big task into smaller, easier-to-handle pieces. So, instead of being intimidated by 100/361, we can think of it as finding the square root of 100 and the square root of 361 individually, which are numbers we can work with more easily.

Solving for x

Okay, let's get down to the nitty-gritty and actually solve for x. Remember our equation: x² = 100/361. The key to unlocking the value of x here is to perform the opposite operation of squaring, which is taking the square root. When we take the square root of both sides of the equation, we're essentially undoing the squaring operation on the left side, leaving us with just x. However, and this is super important, we need to remember that there are two possible solutions: a positive square root and a negative square root. This is because, as we discussed earlier, both a positive and a negative number, when squared, will give you a positive result. So, when we take the square root of 100/361, we need to consider both the positive and negative possibilities. This is often represented mathematically using the plus-minus symbol (±), which indicates that we have two solutions to consider. By keeping this in mind, we won't miss one of the solutions and get the complete picture.

So, we apply the square root to both sides of the equation: √(x²) = ±√(100/361). This simplifies to x = ±√(100/361). Now, here's the neat part about dealing with the square root of a fraction: we can take the square root of the numerator and the denominator separately. This means we can rewrite our equation as x = ±(√100 / √361). This makes the problem much easier to handle because we're now dealing with the square roots of smaller, more familiar numbers. It's like breaking down a complex problem into smaller, more manageable steps. We know that the square root of 100 is 10 because 10 * 10 = 100. And, after a bit of thought (or perhaps remembering some common squares), we can figure out that the square root of 361 is 19 because 19 * 19 = 361. So, we can substitute these values back into our equation.

Putting it all together, we get x = ±(10/19). The ± symbol tells us that we have two solutions: a positive one and a negative one. So, our two solutions for x are x = 10/19 and x = -10/19. These are the two numbers that, when squared, will give us 100/361. We've successfully solved the equation! It's always a good idea to double-check your answers, especially when dealing with square roots. You can do this by plugging each solution back into the original equation and making sure it holds true. For example, if we square 10/19, we get (10/19)² = 100/361, which matches the right side of our original equation. Similarly, if we square -10/19, we also get (-10/19)² = 100/361. This confirms that both solutions are correct. Go us!

Identifying the Correct Answers

Now that we've found the solutions, x = 10/19 and x = -10/19, let's take a look at the options provided and see which ones match. This is a crucial step in problem-solving – making sure we select the answers that we've actually calculated. Sometimes, in multiple-choice questions, there might be distractors, which are answers that look similar but are incorrect. So, it's important to be precise and compare our solutions carefully with the given options. This step ensures that we're not just doing the math correctly but also interpreting the results correctly in the context of the question.

Looking at the options, we need to see if 10/19 and -10/19 are among them. It's possible that the solutions might be presented in a different form, such as a simplified fraction or a decimal. If that's the case, we might need to do a little bit of extra work to see if our solutions are equivalent to any of the options. For example, we might need to simplify a fraction or convert a fraction to a decimal to make the comparison. However, in this case, our solutions are in the simplest form, so we can directly compare them with the options. This is where paying attention to detail really pays off. We want to make sure we're selecting the exact matches and not getting tricked by similar-looking but incorrect answers. Double-checking is always a good practice to ensure accuracy.

After carefully comparing our solutions (x = 10/19 and x = -10/19) with the given options, we can confidently identify the correct answers. If the options include 10/19 and -10/19, then those are the ones we should select. If the options are in a different form, such as decimals, we would need to convert our fractions to decimals to see which ones match. For instance, 10/19 is approximately equal to 0.526, and -10/19 is approximately equal to -0.526. So, if the options were in decimal form, we would look for these values. The key is to be flexible and able to work with numbers in different formats. And that's it! We've successfully solved the equation and identified the correct solutions. High five, guys!

Conclusion

So, there you have it! We've successfully navigated the equation x² = 100/361 and found its solutions. We started by understanding what the equation meant, then we used the concept of square roots to solve for x, remembering to consider both positive and negative solutions. We broke down the problem into manageable steps, taking the square root of the numerator and denominator separately. Finally, we compared our solutions with the given options to identify the correct answers. This process highlights the importance of understanding the underlying concepts, breaking down problems, and double-checking our work. Solving equations like this is a fundamental skill in mathematics, and mastering it opens the door to more complex and exciting mathematical challenges. Keep practicing, and you'll become a math whiz in no time! Remember, math is like a puzzle, and every equation is a new challenge waiting to be solved. So, keep your thinking caps on, and happy solving!