Have you ever stumbled upon a mathematical equation that seemed daunting at first glance? Well, today, we're going to tackle one such equation together: √x-8 = 17. Don't worry, guys, it's not as scary as it looks! We'll break it down step by step, ensuring you not only understand the solution but also the logic behind it. So, let's dive in and demystify this equation.
Understanding the Basics: Square Roots and Equations
Before we jump into solving, let's quickly recap some fundamental concepts. A square root of a number is a value that, when multiplied by itself, gives you the original number. For instance, the square root of 25 is 5 because 5 * 5 = 25. Equations, on the other hand, are mathematical statements asserting the equality of two expressions. Our goal here is to find the value of 'x' that makes the equation √x-8 = 17 true.
The key to solving equations involving square roots is to isolate the square root term and then eliminate it by performing the inverse operation, which in this case is squaring. We need to remember that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the equality. This principle is the cornerstone of solving any algebraic equation, and it's crucial to keep this in mind as we proceed. Think of it like a balancing scale – if you add weight to one side, you must add the same weight to the other to keep it balanced. This ensures that the equation remains valid throughout the solving process.
Furthermore, when dealing with square roots, it's important to consider the domain of the variable. The expression inside the square root (the radicand) must be non-negative, as the square root of a negative number is not a real number. This means that in our equation, x - 8 must be greater than or equal to zero. This consideration will help us determine if our solution is valid or if it's an extraneous solution, which is a solution that arises from the solving process but doesn't satisfy the original equation. So, keep this in mind as we move forward; we'll need to check our final answer to ensure it makes sense in the context of the original equation.
Step-by-Step Solution: √x-8 = 17
Now, let's get our hands dirty and solve this equation! Remember, our aim is to isolate 'x'. Here's how we'll do it:
Step 1: Isolate the Square Root
The first step in solving the equation √x-8 = 17 is to isolate the square root term. In this case, the square root term is already isolated on the left side of the equation, which makes our job a little easier. We don't need to perform any additional operations to get the square root by itself. This is a crucial step because we can't directly eliminate the square root until it's the only term on one side of the equation. Think of it as preparing the ground before planting a seed; we need to clear away any obstacles before we can move on to the next step.
Step 2: Square Both Sides
To eliminate the square root, we need to perform the inverse operation, which is squaring. This means we'll square both sides of the equation. Squaring the left side, (√x-8)², effectively cancels out the square root, leaving us with x - 8. On the right side, we have 17², which is 17 multiplied by itself, resulting in 289. So, our equation now looks like this: x - 8 = 289. This step is vital because it transforms the equation from one involving a square root to a simple linear equation, which is much easier to solve.
The act of squaring both sides is based on the fundamental algebraic principle that if two quantities are equal, then their squares are also equal. This principle allows us to manipulate the equation without changing its fundamental truth. However, it's also important to be mindful that squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. This is why we'll need to check our final answer in the original equation to ensure it's valid.
Step 3: Isolate 'x'
Now that we have a simple linear equation, x - 8 = 289, our next step is to isolate 'x'. To do this, we need to get rid of the -8 on the left side. We can accomplish this by adding 8 to both sides of the equation. This is another application of the balancing principle: what we do to one side, we must do to the other. Adding 8 to both sides gives us x - 8 + 8 = 289 + 8, which simplifies to x = 297. So, we've found a potential solution for 'x'.
This step is a straightforward application of the addition property of equality, which states that adding the same number to both sides of an equation does not change the equality. By adding 8 to both sides, we effectively isolate 'x' on the left side, giving us a clear value for 'x'. However, as we discussed earlier, we still need to verify this solution to ensure it's not an extraneous one.
Step 4: Check the Solution
This is a crucial step that many students overlook, but it's essential to ensure our solution is valid. We need to substitute x = 297 back into the original equation, √x-8 = 17, and see if it holds true. Plugging in 297 for 'x', we get √(297 - 8) = 17. Simplifying inside the square root, we have √289 = 17. The square root of 289 is indeed 17, so we have 17 = 17, which is a true statement. This confirms that x = 297 is a valid solution to the equation.
Checking the solution is not just a formality; it's a critical part of the problem-solving process. It helps us catch any extraneous solutions that might have been introduced when we squared both sides of the equation. By substituting our solution back into the original equation, we're essentially asking, "Does this value of 'x' actually make the equation true?" If the answer is yes, then we have a valid solution. If the answer is no, then we need to discard that solution and re-examine our work.
The Final Answer
After carefully following each step and verifying our solution, we can confidently state that the solution to the equation √x-8 = 17 is x = 297. Congratulations, guys! We've successfully solved this equation together.
Tips for Solving Similar Equations
Now that we've conquered this equation, let's arm ourselves with some tips for tackling similar problems in the future:
- Always Isolate the Square Root: This is the golden rule. Before you do anything else, make sure the square root term is alone on one side of the equation.
- Square Both Sides Carefully: When squaring, pay close attention to any binomials or expressions with multiple terms. Remember to use the distributive property or the FOIL method if necessary.
- Check for Extraneous Solutions: This is non-negotiable. Always, always, always substitute your solution back into the original equation to verify its validity.
- Practice Makes Perfect: The more equations you solve, the more comfortable and confident you'll become. So, keep practicing!
By following these tips and understanding the underlying principles, you'll be well-equipped to solve a wide range of equations involving square roots. Remember, mathematics is a journey, not a destination. Embrace the challenges, learn from your mistakes, and celebrate your successes. You've got this, guys!
Conclusion
Solving equations involving square roots might seem tricky at first, but with a clear understanding of the steps involved and a healthy dose of practice, you can master them. We've walked through the solution to √x-8 = 17, highlighting the importance of isolating the square root, squaring both sides, and, most importantly, checking for extraneous solutions. Remember, guys, mathematics is a skill that improves with practice. So, keep exploring, keep learning, and keep solving! You're on your way to becoming a math whiz!