Solving √(x² + 49) = X + 5 A Step-by-Step Guide

by Pedro Alvarez 48 views

Hey guys! Today, we're diving into a fun little mathematical problem: finding the solution to the equation √(x² + 49) = x + 5. This might look intimidating at first, but don't worry, we'll break it down step by step. Math can be like a puzzle, and we're here to solve it together. We'll explore the nuances of solving equations involving square roots and discuss why certain solutions might appear valid but aren't. By the end of this guide, you'll not only know the correct answer but also understand the process of getting there. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we fully understand the problem. The equation we're dealing with is √(x² + 49) = x + 5. This is a radical equation because it involves a square root. Our goal is to find the value(s) of x that make this equation true. Remember, when dealing with square roots, we need to be cautious about extraneous solutions – values we get during the solving process that don't actually satisfy the original equation. These can pop up because squaring both sides of an equation can sometimes introduce solutions that weren't there initially. To successfully navigate this, we need to understand the underlying principles of algebraic manipulation and the properties of square roots. So, let's first lay out the plan of attack and then execute it with precision. We'll start by isolating the square root and then squaring both sides, a common technique for dealing with radicals. But we won't stop there; we'll meticulously check our solutions to ensure they're legitimate. This attention to detail is what turns a good problem-solver into a great one. So, stick with me, and let's conquer this equation!

Step-by-Step Solution

Okay, let's get our hands dirty and solve this equation step-by-step. Here’s how we can tackle √(x² + 49) = x + 5:

  1. Isolate the Square Root: In this case, the square root is already isolated on the left side of the equation. Awesome! That saves us a step.
  2. Square Both Sides: To get rid of the square root, we'll square both sides of the equation. This gives us: (√(x² + 49))² = (x + 5)². Squaring the left side simply removes the square root, leaving us with x² + 49. On the right side, we need to expand (x + 5)², which means (x + 5) * (x + 5). Remember our algebraic expansion rules! This gives us x² + 10x + 25. So now our equation looks like this: x² + 49 = x² + 10x + 25.
  3. Simplify the Equation: Now, let’s simplify things. We have x² on both sides, so we can subtract x² from both sides, which cancels them out. This leaves us with 49 = 10x + 25. See? It's getting simpler already!
  4. Solve for x: Next, we want to isolate x. Subtract 25 from both sides: 49 - 25 = 10x + 25 - 25, which simplifies to 24 = 10x. Now, divide both sides by 10 to get x by itself: 24 / 10 = x. This simplifies to x = 12/5.
  5. Check for Extraneous Solutions: This is super important! We need to plug our solution, x = 12/5, back into the original equation to make sure it works. Let's do it: √( (12/5)² + 49 ) = 12/5 + 5. First, let's simplify the left side: (12/5)² is 144/25. So we have √( 144/25 + 49 ). To add these, we need a common denominator. 49 is the same as 49/1, which is the same as 1225/25. So we have √( 144/25 + 1225/25 ) = √( 1369/25 ). The square root of 1369 is 37, and the square root of 25 is 5. So the left side simplifies to 37/5. Now, let's look at the right side: 12/5 + 5. Again, we need a common denominator. 5 is the same as 25/5. So we have 12/5 + 25/5 = 37/5. Hooray! The left side (37/5) equals the right side (37/5). This means x = 12/5 is a valid solution.

So, there you have it! We've successfully solved the equation and verified our solution. It’s all about taking it one step at a time and being meticulous with our work.

Why Checking for Extraneous Solutions is Crucial

Guys, I can't stress enough how crucial it is to check for extraneous solutions when you're dealing with radical equations. It's like the golden rule of radical equations! Let's talk a bit more about why this step is so important.

The root of the issue (pun intended!) lies in the act of squaring both sides of an equation. Squaring can introduce solutions that weren't there originally. Think of it this way: if we have two numbers, a and b, and we know that a = b, then it's definitely true that a² = b². However, the reverse isn't always true. If a² = b², it doesn't necessarily mean that a = b. For example, if a = -3 and b = 3, then a² = 9 and b² = 9, so a² = b², but a is not equal to b. This is because squaring eliminates the sign, making both positive and negative values appear the same.

In our equation, √(x² + 49) = x + 5, squaring both sides was a necessary step to get rid of the square root. But it opened the door to potential extraneous solutions. That's why, after we found x = 12/5, we had to plug it back into the original equation. If we hadn't, we might have mistakenly thought we had the correct answer, when in reality, it could have been a false lead.

Checking for extraneous solutions ensures that we're only accepting values that truly satisfy the original equation, the one with the square root intact. It's like a final quality control check in our problem-solving process. It’s a little extra work, yes, but it's the difference between getting the right answer and falling into a trap. So, always remember to check those solutions! It’s a habit that will save you a lot of headaches in the long run.

Common Mistakes to Avoid

Alright, let's chat about some common pitfalls that students often encounter when solving equations like √(x² + 49) = x + 5. Knowing these mistakes beforehand can help you steer clear of them and boost your problem-solving accuracy.

  1. Forgetting to Check for Extraneous Solutions: We’ve hammered this point home, but it’s worth repeating! This is the most common mistake. Students often get to the solution x = 12/5, circle it, and move on, thinking they’re done. But without that crucial check, they could be including a value that doesn't actually work. Always, always, always substitute your solution back into the original equation.
  2. Incorrectly Squaring (x + 5): Remember that (x + 5)² means (x + 5) * (x + 5), not x² + 5². You need to use the distributive property (or the FOIL method) to expand it correctly: (x + 5)(x + 5) = x² + 5x + 5x + 25 = x² + 10x + 25. A simple mistake here can throw off the entire solution.
  3. Algebraic Errors in Simplification: It's easy to make small algebraic slips, especially when dealing with multiple steps. For instance, when simplifying 49 = 10x + 25, a common mistake is to add 25 to both sides instead of subtracting. These little errors can accumulate, leading to a wrong answer. Double-check each step to ensure you're applying the correct operations.
  4. Misunderstanding the Square Root: The square root function, by definition, gives the non-negative root. So, √(9) is 3, not ±3. This is important when checking for extraneous solutions. If you end up with a negative value on the square root side of the equation, you know something’s amiss.
  5. Giving Up Too Early: Sometimes, these problems can seem a bit tricky, and it's tempting to throw in the towel. But don't! Break the problem down into smaller steps, and tackle each step methodically. Math is like a muscle; the more you exercise it, the stronger it gets. So, persevere, and you'll get there!

By being aware of these common mistakes, you can approach radical equations with more confidence and precision. Remember, it's not just about getting the right answer; it's about understanding the process and avoiding the common pitfalls along the way.

Conclusion

So, guys, we've successfully navigated the equation √(x² + 49) = x + 5 and found the solution: x = 12/5. We've not only solved the problem but also delved into the reasons why certain steps are necessary, like checking for extraneous solutions. Math isn't just about memorizing formulas; it's about understanding the underlying logic and principles.

We talked about the importance of checking for extraneous solutions because squaring both sides of an equation can introduce false solutions. We highlighted the common mistakes to avoid, such as incorrectly squaring binomials or making algebraic slips during simplification. And we emphasized the significance of a step-by-step approach, breaking down the problem into manageable chunks.

Remember, solving mathematical problems is a skill that improves with practice. The more you engage with problems like this, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're valuable learning opportunities. The key is to learn from those mistakes and refine your problem-solving techniques.

So, keep practicing, keep exploring, and keep challenging yourselves. Math is a fascinating journey, and we're all in it together. Until next time, happy solving!