Hey guys! Let's dive into a cool math problem today. We're going to break down an equation step-by-step, just like we're solving a puzzle together. Our mission? To figure out what Brianne could have written in Step 2 of her math problem. It's like we're math detectives, piecing together the clues to crack the case. This isn't just about getting the right answer; it's about understanding how we get there, the logic and the steps involved. So, buckle up, math enthusiasts! We're about to embark on a journey through the world of equations, squares, and solutions. Let's get started and see what Brianne was up to!
The Mystery Equation: 3(x + 4)² = 48
So, we've got this equation, 3(x + 4)² = 48, and it's our starting point. It might look a bit intimidating at first glance, but don't worry, we're going to break it down into bite-sized pieces. Think of it as a mathematical treasure map, and each step is a clue that leads us closer to the final answer. The equation involves a variable, x, some parentheses, a square, and a few numbers. Our goal is to figure out what values of x make this equation true. In other words, we're trying to find the 'x' that balances the equation, making both sides equal. This is the core of algebra – solving for the unknown, and it's a fundamental skill in mathematics. The beauty of algebra lies in its systematic approach, and we're about to see that in action as we follow Brianne's steps.
Step 1: (x + 4)² = 16
Brianne's first move, (x + 4)² = 16, gives us a major clue. To get here, she had to simplify the original equation. Can you guess what she did? If you're thinking she divided both sides of the original equation by 3, you're spot on! This is a classic algebraic technique – doing the same thing to both sides of the equation to keep it balanced. It's like a seesaw; if you add or remove weight from one side, you have to do the same on the other to keep it level. Dividing by 3 isolates the squared term, (x + 4)², which makes the equation much easier to work with. This step is crucial because it sets the stage for solving for x. We've now transformed our original equation into a simpler form, and that's a big win in the world of math. It's like clearing away the underbrush in a forest to get a better view of the path ahead.
The Critical Step 2: What Could Brianne Have Written?
Now, this is where the fun begins! We need to figure out what Brianne could have written as Step 2. She jumped from (x + 4)² = 16 to the solutions x = -8 or x = 0 in Step 3. That's quite a leap, so there must be an intermediate step she took. The key here is to understand what it means to have something squared equal to 16. Think about it: what numbers, when multiplied by themselves, give you 16? You'll probably come up with 4, because 4 * 4 = 16. But there's another number too! Remember that negative numbers, when squared, become positive. So, (-4) * (-4) also equals 16. This is super important because it means that (x + 4) could be either 4 or -4. This realization is the bridge between Step 1 and Step 3, and it's the core of what Brianne likely did in Step 2. So, let's formulate that mathematically and see what it looks like.
Brianne's Step 2 most likely involved taking the square root of both sides of the equation. When you do this, you have to remember that there are two possibilities: the positive square root and the negative square root. So, the equation (x + 4)² = 16 becomes:
x + 4 = ±4
This is the most logical step because it directly addresses the square in the equation. It's like peeling back the layers of an onion; we're undoing the operations to get closer to x. The ± symbol is crucial here. It's a shorthand way of saying "plus or minus," and it reminds us that there are two possible values for the square root of 16. Without considering both the positive and negative roots, we'd miss one of the solutions for x. This step is a perfect example of how mathematical notation can be efficient and precise. It packs a lot of information into a small space, and it's essential for solving equations like this. So, Brianne's Step 2 was almost certainly x + 4 = ±4, and this sets us up perfectly for finding the two possible values of x.
Step 3: x = -8 or x = 0
Alright, let's break down Step 3: x = -8 or x = 0. This is the grand finale, where we actually find the solutions for x. Brianne has told us the answers, but let's make sure we understand how she got there from Step 2 (which we figured out was x + 4 = ±4). Remember that ± symbol? It means we have two separate little equations to solve:
- x + 4 = 4
- x + 4 = -4
Let's tackle the first one. To solve x + 4 = 4, we need to isolate x. The way we do that is by subtracting 4 from both sides of the equation. This gives us x = 4 - 4, which simplifies to x = 0. So, there's one of our solutions! Now, let's move on to the second equation, x + 4 = -4. Again, we want to get x by itself, so we subtract 4 from both sides. This gives us x = -4 - 4, which simplifies to x = -8. And there's our second solution! We've found both values of x that make the original equation true. It's like we've completed the puzzle, fitting all the pieces together perfectly. These solutions are the culmination of all our hard work, and they demonstrate the power of algebra to solve for the unknown.
Conclusion: Cracking the Code with Brianne
So, guys, we've done it! We successfully reconstructed Brianne's steps and figured out what she likely wrote in Step 2. By understanding the underlying principles of algebra, like balancing equations and considering both positive and negative roots, we were able to fill in the missing piece of the puzzle. Remember, math isn't just about memorizing formulas; it's about understanding the why behind the how. By breaking down the problem step-by-step, we not only found the answer but also gained a deeper appreciation for the process. Next time you encounter a tricky equation, remember Brianne's steps and our detective work. You've got the tools to solve it! Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!