Expanding (√2 - 3√8)²: A Step-by-Step Guide

Hey guys! Let's dive into a common algebra problem: expanding the expression (√2 - 3√8)². This might seem a bit intimidating at first, but don't worry, we'll break it down step by step. By the end of this guide, you'll not only know how to solve this specific problem but also understand the underlying principles so you can tackle similar challenges with confidence. We will explore the fundamental concepts needed to expand this expression, including the binomial square formula and simplification of radicals. So, grab your pencils and let's get started!

Understanding the Basics

Before we jump into the expansion, let's refresh some key concepts. First, remember the binomial square formula: (a - b)² = a² - 2ab + b². This formula is crucial for expanding expressions in the form of a square of a difference. In our case, a will be √2 and b will be 3√8. Next, we need to understand how to simplify radicals. A radical is simply a root, like the square root (√). Simplifying radicals often involves finding perfect square factors within the radicand (the number under the root). For instance, √8 can be simplified because 8 has a perfect square factor of 4 (8 = 4 × 2). Knowing these basics will make the expansion process much smoother. Think of these as the essential tools in your mathematical toolkit – you can't build anything sturdy without them! So, let's keep these concepts in mind as we move forward. Mastering these fundamental concepts will not only help you solve this particular problem but also equip you with a solid foundation for tackling more complex algebraic expressions in the future. Practice applying these principles to various problems to solidify your understanding. This will boost your confidence and make algebra much less daunting.

Step 1: Applying the Binomial Square Formula

Alright, let's get our hands dirty and start expanding (√2 - 3√8)². Using the binomial square formula, (a - b)² = a² - 2ab + b², we can substitute a with √2 and b with 3√8. This gives us: (√2 - 3√8)² = (√2)² - 2(√2)(3√8) + (3√8)². Now, let's break this down piece by piece. First, we have (√2)², which is simply 2 because the square of a square root cancels out the root. Next, we have -2(√2)(3√8). Let's leave this for a moment and move on to the last term, (3√8)². Remember, when you square a product, you square each factor. So, (3√8)² = 3² × (√8)² = 9 × 8 = 72. Now, let's revisit the middle term, -2(√2)(3√8). We can rewrite this as -6(√2)(√8). When multiplying radicals, you can multiply the numbers inside the square roots, so this becomes -6√(2 × 8) = -6√16. And since √16 is 4, the middle term simplifies to -6 × 4 = -24. So, putting it all together, we have: (√2)² - 2(√2)(3√8) + (3√8)² = 2 - 24 + 72. See? It's all coming together! The key here is to take it one step at a time, applying the formula methodically and simplifying each term as you go. This approach prevents errors and makes the whole process much more manageable. Remember, math is like building with Lego bricks – each step is a piece that fits together to create the final structure.

Step 2: Simplifying the Expression

Okay, we've expanded the expression and now we have 2 - 24 + 72. The next step is to simplify this down to a single number. This part is relatively straightforward, but it's still important to be careful with your arithmetic to avoid any silly mistakes. Let's combine the numbers step by step. First, we can combine 2 and -24, which gives us -22. So, our expression now looks like -22 + 72. Next, we add -22 and 72. Think of it as 72 - 22, which equals 50. Therefore, the simplified form of (√2 - 3√8)² is 50. Woo-hoo! We're almost there! You see, simplifying is like cleaning up after a good cooking session – you've done the hard work of preparing the meal (expanding the expression), and now you're just tidying things up to see the delicious result. Always double-check your calculations, especially when dealing with negative numbers, to ensure accuracy. A small error in arithmetic can throw off the whole result, so it's worth taking that extra moment to be sure. This step highlights the importance of basic arithmetic skills in algebra. Even the most complex algebraic problems often boil down to simple addition, subtraction, multiplication, and division. So, keep practicing those fundamental skills!

Step 3: Alternative Approach by Simplifying Radicals First

Now, before we wrap things up, let's explore an alternative approach to solving this problem. Sometimes, simplifying radicals before expanding can make the process easier. Remember our original expression: (√2 - 3√8)². Notice that √8 can be simplified. Since 8 = 4 × 2, we can write √8 as √(4 × 2). And because √4 = 2, we have √8 = 2√2. Now, let's substitute this back into our original expression: (√2 - 3√8)² becomes (√2 - 3(2√2))². This simplifies to (√2 - 6√2)². See how we've made things a bit cleaner already? Next, we can combine the terms inside the parentheses. Think of √2 as 1√2. So, we have 1√2 - 6√2, which is -5√2. Now our expression looks like (-5√2)². This is much simpler to expand! Squaring -5√2 means squaring both -5 and √2. So, (-5√2)² = (-5)² × (√2)² = 25 × 2 = 50. And there you have it – the same answer, but a slightly different route. This alternative approach highlights the importance of flexibility in problem-solving. Sometimes, there isn't just one "right" way to tackle a math problem. Exploring different strategies can lead you to a solution more efficiently and deepen your understanding of the underlying concepts. In this case, simplifying the radical first reduced the complexity of the terms we were working with, making the expansion process smoother. It's like choosing the right tool for the job – sometimes a wrench is better than a screwdriver, and vice versa. So, always be open to trying different approaches and see what works best for you.

Conclusion: Mastering Expansion and Simplification

Alright, guys, we've reached the end of our journey to expand (√2 - 3√8)². We not only found the answer, which is 50, but also explored the essential concepts and techniques involved. We started with the basics, like the binomial square formula and simplifying radicals, and then we applied these principles step-by-step. We even looked at an alternative approach by simplifying radicals first. This problem illustrates the importance of a solid foundation in algebra. Understanding fundamental concepts like the binomial square formula and how to simplify radicals is crucial for tackling more complex problems. But beyond the mechanics of the solution, we've also touched on the importance of problem-solving strategies. There's often more than one way to skin a cat, and being flexible and open to different approaches can make your mathematical journey much more rewarding. Remember, practice makes perfect. The more you work through problems like this, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and never be afraid to try new things. Math is like a puzzle – sometimes it takes a bit of trial and error to find the right fit, but the satisfaction of solving it is well worth the effort. You've got this! Now go out there and conquer those algebraic expressions!