Solving √(y+6)+2=√(y+26) Why Subtracting 2 Is The First Step

Hey guys! Let's dive into solving the radical equation √(y+6) + 2 = √(y+26). If you've ever stared at a math problem and wondered where to even begin, you're definitely not alone. Mathway suggests kicking things off by subtracting 2 from both sides, and that might seem a bit out of the blue. But trust me, there’s a solid reason behind this move. Think of solving equations like planning a strategic game. The goal is to get the variable, in this case, y, all by its lonesome on one side of the equation. To do that, we need to undo all the operations that are messing with it. When you see square roots, the game plan usually involves isolating them and then squaring both sides to get rid of those pesky radicals. So, why subtract 2 first? Well, it’s all about setting ourselves up for success in the next steps. By getting that constant term out of the way, we make it easier to isolate the square root and move forward. Radical equations can seem intimidating, but they’re really just puzzles waiting to be solved. The key is to break them down into manageable steps and understand the logic behind each move. We'll walk through this problem together, step by step, so you can see exactly why this approach works and how you can apply it to other similar problems. Stick with me, and you'll be tackling radical equations like a pro in no time!

The Strategy Behind Subtracting 2

When we are faced with the equation √(y+6) + 2 = √(y+26), the initial step of subtracting 2 from both sides might seem a bit mysterious, but it's actually a clever move rooted in algebraic strategy. The primary goal in solving any equation is to isolate the variable, which in this case is y. However, y is currently trapped inside a square root, making it difficult to directly manipulate. So, how do we free it? The golden rule when dealing with radicals is to isolate them before squaring. Squaring both sides of an equation is a powerful tool for eliminating square roots, but it works best when the radical term is by itself. If we were to square the original equation right away, we'd end up with a much more complicated mess due to the presence of the +2 term. Remember the binomial square formula: (a + b)² = a² + 2ab + b². Squaring the left side as is would introduce a cross term (2 * √(y+6) * 2), which would keep a radical in the equation and make things even more tangled. By subtracting 2 from both sides first, we strategically isolate one of the square root terms. This gives us a cleaner equation to work with: √(y+6) = √(y+26) - 2. Now, when we square both sides, we'll still have some work to do, but the equation will be significantly simpler to manage. Think of it like clearing a path through a dense forest. Subtracting 2 is like taking the first step to remove some of the underbrush, making it easier to see the path ahead. This step is not just about blindly following a rule; it's about setting up the problem in the most advantageous way possible. It’s a tactical decision that streamlines the solving process and prevents unnecessary complications. So, by understanding this strategy, you're not just learning how to solve this specific equation, you're learning a valuable problem-solving technique that you can apply to many other radical equations. It's all about making the right moves in the right order to reach the solution efficiently. This initial subtraction is a key move in our strategic game, guiding us toward a clearer and more manageable equation.

Step-by-Step Solution Walkthrough

Let's break down the step-by-step solution to the equation √(y+6) + 2 = √(y+26), so you can see exactly how and why each move is made. This will help solidify your understanding and give you the confidence to tackle similar problems. First things first, we start with our original equation: √(y+6) + 2 = √(y+26). As Mathway suggests, our initial move is to subtract 2 from both sides. This is a strategic step to isolate one of the square root terms, which we discussed earlier. Subtracting 2 from both sides gives us: √(y+6) = √(y+26) - 2. Now that we have a square root term isolated, the next logical step is to eliminate the square root. We do this by squaring both sides of the equation. Squaring both sides gives us: (√(y+6))² = (√(y+26) - 2)². On the left side, the square root and the square cancel each other out, leaving us with y + 6. On the right side, we need to carefully expand the binomial. Remember the formula (a - b)² = a² - 2ab + b². Applying this formula, we get: (√(y+26) - 2)² = (√(y+26))² - 2 * √(y+26) * 2 + 2² = y + 26 - 4√(y+26) + 4. So, our equation now looks like this: y + 6 = y + 26 - 4√(y+26) + 4. Notice that we still have a square root in the equation, but we've made progress. Let's simplify and isolate the remaining radical term. First, we can subtract y from both sides, which cancels out the y terms: 6 = 26 - 4√(y+26) + 4. Combine the constants on the right side: 6 = 30 - 4√(y+26). Now, let's isolate the radical term by subtracting 30 from both sides: 6 - 30 = -4√(y+26), which simplifies to -24 = -4√(y+26). Divide both sides by -4 to further isolate the square root: (-24) / (-4) = √(y+26), which gives us 6 = √(y+26). We're almost there! To get rid of the square root, we square both sides again: 6² = (√(y+26))², which simplifies to 36 = y + 26. Finally, subtract 26 from both sides to solve for y: 36 - 26 = y, so y = 10. But wait, we're not quite done yet! It’s crucial to check our solution in the original equation to make sure it’s valid. Plug y = 10 back into the original equation: √(10+6) + 2 = √(10+26). This simplifies to √16 + 2 = √36, which further simplifies to 4 + 2 = 6, and indeed, 6 = 6. Our solution checks out! So, the final answer is y = 10. By walking through each step methodically, we've successfully solved the radical equation. Remember, the key is to isolate the radical, square both sides, and then solve for the variable. And always, always check your solution! Solving radical equations might seem like a complex dance, but with practice and a clear understanding of each step, you can master it. Keep practicing, and you’ll find these problems become much easier to handle.

Why This Approach Works

Let's delve deeper into why this particular approach—subtracting 2 from both sides at the beginning of the equation √(y+6) + 2 = √(y+26)—works so effectively. It's not just a random step; it’s a carefully chosen tactic that streamlines the entire problem-solving process. The underlying principle here is strategic simplification. In mathematics, especially when dealing with complex equations, the key to success often lies in breaking down the problem into smaller, more manageable parts. By subtracting 2, we’re essentially clearing a pathway to isolate the square root terms, which are the trickiest part of this equation. Think of it like untangling a knot. You wouldn't yank on it haphazardly; instead, you'd try to loosen the individual strands one at a time. Similarly, in this equation, we're trying to