Solve X^2 - 100 = 0 By Square Roots

Hey math whizzes! Today, we're diving into a super straightforward method for solving quadratic equations: solving by taking square roots. It's a technique that's perfect for equations that are missing the 'x' term, like our example: $x^2 - 100 = 0$. This method is all about isolating the squared term and then, you guessed it, taking the square root of both sides to find those elusive 'x' values. It's a fundamental skill in algebra, and once you get the hang of it, you'll be zipping through problems like a pro. So, grab your calculators, get comfy, and let's break down how to tackle this equation and find the correct answer from the given options.

Understanding the Method: Solving by Taking Square Roots

Alright guys, let's talk about solving quadratic equations by taking square roots. This method is your best friend when you have an equation in the form of $ax^2 + c = 0$, meaning there's no 'bx' term. The core idea is simple: get that $x^2$ term all by itself on one side of the equation, and then take the square root of both sides. Easy peasy, right? But here's a crucial detail that many people forget: when you take the square root of both sides, you need to remember that there are two possible solutions – a positive one and a negative one. This is because squaring a positive number and squaring its negative counterpart both result in a positive number. For instance, $(5)^2 = 25$ and $(-5)^2 = 25$. So, if we know $x^2 = 25$, then 'x' could be either 5 or -5. This is why we often use the plus-minus symbol, $\pm$, to represent both possibilities. In our specific problem, $x^2 - 100 = 0$, we'll need to perform a couple of steps to isolate $x^2$. First, we'll add 100 to both sides to get $x^2$ alone. Then, we'll take the square root of both sides, making sure to account for both the positive and negative roots. This systematic approach ensures we don't miss any potential solutions, which is super important in mathematics. It's a foundational technique that opens the door to understanding more complex quadratic concepts later on, so mastering it now will pay dividends as you continue your math journey. We'll explore the specific steps for $x^2 - 100 = 0$ in the next section, making sure you guys can follow along and nail this type of problem every time. The beauty of this method lies in its directness; it cuts through the complexity when applicable, allowing for quick and accurate solutions.

Step-by-Step Solution for $x^2 - 100 = 0$

Let's get down to business and solve $x^2 - 100 = 0$ using our solving by taking square roots technique. Follow these steps, and you'll see how straightforward it is. First off, we need to isolate the $x^2$ term. Right now, it's sitting there with '-100' next to it. To get rid of that '-100', we do the opposite operation, which is adding 100 to both sides of the equation. So, we have: $x^2 - 100 + 100 = 0 + 100$. This simplifies to $x^2 = 100$. Now that $x^2$ is all by itself, we can move on to the next crucial step: taking the square root of both sides. Remember what we talked about? We need to consider both the positive and negative roots. So, when we take the square root of $x^2$, we get $x$. And when we take the square root of 100, we get both positive 10 and negative 10. Mathematically, we write this as $x = \pm\sqrt{100}$. Since the square root of 100 is 10, our solution becomes $x = \pm 10$. This means there are two possible values for 'x' that satisfy the original equation: $x = 10$ and $x = -10$. To double-check our work, we can plug these values back into the original equation. If $x=10$, then $(10)^2 - 100 = 100 - 100 = 0$. Perfect! If $x=-10$, then $(-10)^2 - 100 = 100 - 100 = 0$. Also perfect! Both values work. This confirms that our solving by taking square roots method gave us the correct answers. It's a clean and efficient way to solve this type of quadratic equation. The key takeaway here is to always remember the $\pm$ when you take the square root of both sides of an equation to isolate a squared variable. Missing that detail can lead to an incomplete solution, which we definitely want to avoid in math, right?

Analyzing the Options: Finding the Correct Answer

Now that we've meticulously solved the equation $x^2 - 100 = 0$ and found our solutions, let's take a look at the multiple-choice options provided. Our goal is to identify which option accurately reflects our findings from the solving by taking square roots method. We determined that the solutions for $x$ are $x = 10$ and $x = -10$. This is commonly expressed using the plus-minus notation as $\pm 10$. Let's examine each option:

• A. 10: This option only provides the positive solution. While 10 is a solution, it's not the complete set of solutions for the equation $x^2 - 100 = 0$. Therefore, this is not the best answer.
• B. $\pm 10$: This option includes both the positive solution (10) and the negative solution (-10). This perfectly matches our derived solutions from solving by taking square roots. It acknowledges that squaring either 10 or -10 results in 100, thus satisfying the equation $x^2 = 100$. This looks like our winner, guys!
• C. $\pm 1001$: This option presents $\pm 1001$. Let's quickly check if this is even close. If $x = 1001$, then $x^2 = 1001^2 = 1,002,001$. Clearly, $1,002,001 - 100 eq 0$. Even if we consider $-1001$, its square is also $1,002,001$. So, this option is way off base and doesn't satisfy our equation at all.
• D. 100: This option provides only the value 100. If we substitute $x = 100$ into the equation, we get $(100)^2 - 100 = 10,000 - 100 = 9,900$, which is definitely not 0. This option is also incorrect.

Based on our step-by-step analysis and the crucial understanding of solving by taking square roots that includes both positive and negative solutions, option B, $\pm 10$, is the only correct answer. It accurately represents all the values of $x$ that make the equation $x^2 - 100 = 0$ true. Remember, always consider both roots when applying this method!

Why Understanding Square Roots is Key

So, why is solving by taking square roots such a fundamental skill, and why is it important to grasp the concept of both positive and negative roots? In mathematics, especially when dealing with equations, accuracy and completeness are paramount. When you encounter an equation like $x^2 = k$ (where $k$ is a positive number), you're essentially asking, "What number, when multiplied by itself, equals $k$?" The beauty and sometimes the trickiness of squares is that both a positive number and its negative counterpart, when squared, yield the same positive result. For example, $7 imes 7 = 49$, and $(-7) imes (-7) = 49$. If your equation simplifies to $x^2 = 49$, then 'x' could be either 7 or -7. If you only reported $x=7$, you'd be missing half the picture! This is why the $\pm$ symbol is so indispensable. It's a concise way to say, "The solution is this number and its opposite." In the context of our problem, $x^2 - 100 = 0$, which led us to $x^2 = 100$, failing to include both 10 and -10 would mean we haven't fully solved the equation. This principle extends beyond simple algebra. In geometry, for instance, lengths are typically positive, but in other areas of math and physics, negative values have real-world significance. Understanding the dual nature of square roots helps build a robust mathematical foundation. It prepares you for more advanced topics where the sign of a variable or a solution can drastically change the interpretation of results. So, when you're solving by taking square roots, always pause and ask yourself, "Did I remember the $\pm$?" It's a small detail that makes a massive difference in getting the correct and complete answer. Mastering this ensures you're not just memorizing steps but truly understanding the underlying mathematical principles, which is what Plastik Magazine is all about – giving you the knowledge to excel!

Conclusion: Mastering $x^2 - 100 = 0$ and Beyond

In conclusion, guys, we've successfully tackled the equation $x^2 - 100 = 0$ using the powerful and efficient method of solving by taking square roots. We broke down the process step-by-step: first, isolating the $x^2$ term by adding 100 to both sides, which gave us $x^2 = 100$. Then, we carefully took the square root of both sides, remembering to include both the positive and negative possibilities, leading us to $x = \pm 10$. We then analyzed the given multiple-choice options and confirmed that B. $\pm 10$ is the only correct answer because it encompasses both solutions that satisfy the original equation. This exercise highlights the critical importance of remembering the $\pm$ sign when taking square roots in algebraic equations; it’s a detail that ensures you find all the solutions. The method of solving by taking square roots is a foundational technique for quadratic equations that lack a linear (x) term, and its mastery will serve you well as you progress in your math studies. Keep practicing these types of problems, always double-checking your work by plugging your solutions back into the original equation. Understanding these core concepts is what truly builds mathematical confidence and competence. So, next time you see an equation like this, you’ll know exactly how to approach it and confidently select the right answer. Keep those calculators handy and your minds sharp!