Solving √x - 9 = -6 A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumble upon an equation that looks like a tangled mess? Well, today we're going to grab our mathematical scissors and carefully cut through the knots of the equation √x - 9 = -6. Don't worry, it's not as scary as it seems! We'll break it down step-by-step, so you can confidently solve similar problems in the future. Let's dive in!

Understanding the Equation

At first glance, this equation, √x - 9 = -6, might seem intimidating, but let's break it down. The key here is understanding what each part represents. The symbol '√' is the square root symbol, which means we're looking for a number that, when multiplied by itself, equals 'x'. The 'x' is our unknown variable – the mystery we're trying to solve. The numbers -9 and -6 are constants, meaning they have a fixed value. Our goal is to isolate 'x' on one side of the equation so we can uncover its value. Think of it like a detective solving a case, we're gathering clues and using logic to find the culprit, which in this case is the value of 'x'. The first step in unraveling this equation is to isolate the square root term. We want to get the '√x' by itself on one side of the equation. To do this, we need to get rid of the '-9' that's hanging around. Remember the golden rule of equations: what you do to one side, you must do to the other. So, to get rid of the '-9', we'll add 9 to both sides of the equation. This might seem like a simple step, but it's crucial for setting us up for success. We are essentially undoing the subtraction operation, bringing us closer to isolating our variable. Many students find it helpful to visualize this process. Imagine the equation as a balanced scale. To keep the scale balanced, any operation we perform on one side must also be performed on the other. Adding 9 to both sides maintains this balance, ensuring that the equation remains valid. Understanding this fundamental principle is key to mastering equation solving.

Isolating the Square Root

So, let's put this into action! We start with our original equation: √x - 9 = -6. Now, we're going to add 9 to both sides, as we discussed. This gives us: √x - 9 + 9 = -6 + 9. On the left side, the -9 and +9 cancel each other out, leaving us with just √x. On the right side, -6 + 9 equals 3. So, our equation now looks much simpler: √x = 3. We've successfully isolated the square root! This is a major milestone. We've transformed a slightly complex equation into a much more manageable one. By isolating the square root, we've essentially cleared the path to finding the value of 'x'. It's like we've zoomed in on the key part of the problem, making it easier to see the solution. Now, let's think about what this equation is telling us. It says that the square root of some number 'x' is equal to 3. In other words, we're looking for a number that, when you take its square root, you get 3. This is a classic square root problem, and we're well on our way to solving it. Remember, math is like building with blocks. Each step builds upon the previous one, and by carefully following the steps, we can construct the solution. Isolating the square root was a crucial block in our mathematical construction.

Squaring Both Sides

Now, we're at the exciting part! We have √x = 3. To get rid of the square root, we need to do the opposite operation: squaring. Remember, the square of a number is just that number multiplied by itself (e.g., 3 squared is 3 * 3 = 9). The key here is that squaring a square root cancels it out, leaving us with just the number inside the square root. But, like before, we need to do the same thing to both sides of the equation to keep it balanced. So, we'll square both sides: (√x)² = 3². This step is like unlocking a secret door. By squaring both sides, we're essentially undoing the square root operation, revealing the value of 'x'. On the left side, (√x)² simply becomes x. This is because the square and the square root are inverse operations, they cancel each other out. On the right side, 3² is 3 multiplied by itself, which equals 9. So, our equation now looks incredibly simple: x = 9. We've done it! We've solved for 'x'! This is the moment of truth, the culmination of all our hard work. We've successfully navigated the equation and found the value of our unknown variable. This step highlights the power of inverse operations in solving equations. By using the opposite operation, we can systematically isolate the variable and find its value. Squaring both sides is a technique that's widely used in algebra, so mastering it is a valuable skill.

The Solution

So, after all that mathematical maneuvering, we've arrived at our answer: x = 9. That means the solution to the equation √x - 9 = -6 is x equals 9. Pretty neat, huh? Let's double-check our work to make sure we're spot-on. To verify our solution, we'll substitute x = 9 back into the original equation: √x - 9 = -6. Replacing 'x' with 9, we get: √9 - 9 = -6. The square root of 9 is 3, so we have: 3 - 9 = -6. And indeed, 3 - 9 does equal -6. So, our solution is correct! This verification step is crucial. It's like the final seal of approval, confirming that we've solved the problem accurately. By plugging our solution back into the original equation, we can ensure that it satisfies the equation's conditions. This process helps to catch any potential errors and gives us confidence in our answer. In the world of mathematics, accuracy is paramount. Double-checking our work is a habit that all successful problem-solvers cultivate. It's a way of ensuring that we're not just going through the motions, but truly understanding the solution and its implications. Now that we've found and verified our solution, we can confidently say that x = 9 is the answer!

Why Other Options Are Incorrect

Let's quickly examine why the other answer choices provided (A. x=3, C. x=√3, and D. x=√15) are incorrect. This is a great way to solidify our understanding and prevent future mistakes. If we substitute x = 3 into the original equation, we get: √3 - 9 = -6. The square root of 3 is approximately 1.73, so 1. 73 - 9 is definitely not equal to -6. So, option A is incorrect. Now, let's try option C, x = √3. Substituting this into the equation, we get: √(√3) - 9 = -6. The square root of the square root of 3 is a decimal value much smaller than 9, so subtracting 9 will result in a number much less than -6. Thus, option C is also incorrect. Finally, let's consider option D, x = √15. Substituting this into the equation gives us: √(√15) - 9 = -6. Similar to option C, the square root of the square root of 15 is a decimal value, and subtracting 9 will not result in -6. So, option D is incorrect as well. By systematically checking the incorrect options, we reinforce our understanding of the problem and the correct solution. This process also helps us to identify common errors and misconceptions that might arise when solving similar equations. It's like we're building a fortress of knowledge, making our understanding stronger and more resilient. Understanding why incorrect answers are wrong is just as important as understanding why the correct answer is right. It provides a deeper and more comprehensive grasp of the concepts involved.

Conclusion

And there you have it! We've successfully navigated the equation √x - 9 = -6 and found the solution: x = 9. We took it step-by-step, isolating the square root, squaring both sides, and verifying our answer. Remember, solving equations is like a puzzle, and each step is a piece that brings us closer to the final picture. By understanding the fundamental principles and practicing regularly, you can become a math-solving pro! So, the next time you encounter an equation that looks a bit daunting, don't fret! Just break it down, follow the steps, and you'll uncover the solution in no time. Keep practicing, keep exploring, and keep having fun with math! You've got this! Remember, the key to success in math is to approach each problem with a positive attitude and a willingness to learn. Every equation is an opportunity to expand your knowledge and sharpen your skills. So, embrace the challenge, and enjoy the journey of mathematical discovery!