Solve √m+15 - 1 = 9: A Step-by-Step Solution

Hey guys! Today, we're diving into a fun math problem that involves solving an equation with a square root. Don't worry, it's not as scary as it looks! We'll break it down step-by-step so you can conquer these types of problems with confidence. Our mission? To solve the equation √m+15 - 1 = 9 and find the value of 'm'. Buckle up, mathletes, let's get started!

Understanding the Equation: Laying the Foundation

Before we jump into solving, let's make sure we understand what the equation is telling us. We have √m+15 - 1 = 9. This equation essentially says, "If you take a number 'm', add 15 to it, then find the square root of the result, and finally subtract 1, you'll end up with 9." Our job is to find the mystery number 'm' that makes this statement true. To solve this, we need to isolate 'm' on one side of the equation. This means getting rid of all the other numbers and operations around it, one by one. We'll do this by performing inverse operations – that is, operations that undo each other. For instance, to get rid of subtraction, we'll use addition; to get rid of a square root, we'll use squaring. Remember, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. Think of it like a scale – if you add weight to one side, you need to add the same weight to the other side to keep it level.

Moreover, it's important to keep in mind the order of operations (PEMDAS/BODMAS) when solving equations. However, when we are solving an equation, we essentially reverse the order of operations. This means we deal with addition and subtraction first, then multiplication and division, and finally exponents and roots. This approach helps us systematically peel away the layers surrounding our variable and get closer to isolating it. Understanding these fundamental concepts is crucial for tackling not just this equation, but any algebraic problem that comes your way. So, let's keep these principles in mind as we move forward with solving for 'm'. We'll take it one step at a time, making sure to explain each operation and why we're doing it. By the end of this guide, you'll have a solid understanding of how to solve equations with square roots and feel much more confident in your algebra skills. Let's continue our journey and see how we can isolate that 'm'!

Step 1: Isolating the Square Root – The First Move

Alright, our first step in solving √m+15 - 1 = 9 is to isolate the square root. This means we want to get the term √m+15 all by itself on one side of the equation. Currently, we have a '-1' hanging out next to it, which we need to get rid of. To do this, we'll use the inverse operation of subtraction, which is addition. We'll add 1 to both sides of the equation. This is super important – whatever we do to one side, we have to do to the other to maintain the balance of the equation. So, let's add 1 to both sides:

√m+15 - 1 + 1 = 9 + 1

On the left side, the '-1' and '+1' cancel each other out, leaving us with just the square root term. On the right side, 9 + 1 equals 10. So, our equation now looks like this:

√m+15 = 10

Awesome! We've successfully isolated the square root. Now, the equation is much simpler and we're one step closer to finding 'm'. This step is crucial because it sets us up for the next operation, which will help us eliminate the square root altogether. Isolating the radical is a common first step in solving equations involving radicals, so mastering this technique is a valuable skill. By focusing on getting the square root term by itself, we simplify the equation and make it easier to work with. This methodical approach is key to solving more complex algebraic problems as well. So, remember, when you see a square root in an equation, your first goal should be to isolate it. With this step under our belt, we're ready to move on to the next challenge: eliminating the square root and finally uncovering the value of 'm'. Let's keep up the momentum!

Step 2: Eliminating the Square Root – Squaring Both Sides

Okay, we've made great progress! We've isolated the square root, and our equation now looks like √m+15 = 10. The next hurdle is to get rid of that pesky square root symbol. How do we do that? By using the inverse operation of taking a square root, which is squaring. Remember, the square root of a number, when squared, gives you the original number back. So, if we square both sides of the equation, we can eliminate the square root on the left side.

Let's square both sides:

(√m+15)² = 10²

On the left side, squaring the square root cancels it out, leaving us with just m+15. On the right side, 10 squared (10²) is 10 * 10, which equals 100. So, our equation now looks like this:

m + 15 = 100

Fantastic! We've successfully eliminated the square root and simplified the equation even further. We're now dealing with a simple linear equation, which is much easier to solve. This step highlights the power of using inverse operations to undo mathematical operations. By squaring both sides, we effectively reversed the square root operation and brought us closer to isolating 'm'. Remember, squaring both sides is a crucial technique when dealing with square roots in equations, and it's a tool you'll use frequently in algebra. It's important to understand why this works – squaring a square root essentially cancels out the root, leaving you with the expression inside the square root. Now that we have a much simpler equation, we're just one step away from finding the value of 'm'. Let's move on to the final step and solve for 'm'!

Step 3: Isolating 'm' – The Final Showdown

Alright, we're in the home stretch! Our equation currently stands at m + 15 = 100. We're so close to finding 'm', we can almost taste it! To isolate 'm', we need to get rid of the '+15' that's hanging out with it. Just like before, we'll use the inverse operation – in this case, subtraction. We'll subtract 15 from both sides of the equation to keep things balanced.

Let's subtract 15 from both sides:

m + 15 - 15 = 100 - 15

On the left side, the '+15' and '-15' cancel each other out, leaving us with just 'm'. On the right side, 100 - 15 equals 85. So, our equation now looks like this:

m = 85

Eureka! We've done it! We've successfully solved for 'm'. The value of 'm' that makes the original equation true is 85. This final step demonstrates the importance of systematically isolating the variable we're trying to solve for. By using inverse operations, we carefully peeled away all the other terms and operations surrounding 'm' until it was all alone on one side of the equation. This is a fundamental technique in algebra, and it's the key to solving all sorts of equations. Remember, the goal is always to isolate the variable, and you can do that by using inverse operations and keeping the equation balanced. Now that we've found the value of 'm', it's always a good idea to check our answer to make sure it's correct. Let's plug 'm = 85' back into the original equation and see if it holds true.

Checking Our Solution – Ensuring Accuracy

Before we celebrate our victory, it's always a smart move to check our solution. This helps us ensure that we haven't made any mistakes along the way and that our answer is indeed correct. We found that m = 85, so let's plug that back into the original equation: √m+15 - 1 = 9

Substituting m = 85, we get:

√85+15 - 1 = 9

Let's simplify step-by-step:

√100 - 1 = 9

The square root of 100 is 10, so:

10 - 1 = 9

And finally:

9 = 9

Woohoo! Our solution checks out! The left side of the equation equals the right side, which means our answer, m = 85, is correct. This step of checking our solution is super important because it gives us confidence in our answer and helps us catch any errors we might have made. It's a good habit to get into, especially when dealing with more complex equations. By plugging our solution back into the original equation, we're essentially verifying that it satisfies the equation's conditions. If the equation holds true, we know we've found the correct answer. If it doesn't, we know we need to go back and review our steps to find the mistake. So, always remember to check your solutions – it's the final piece of the puzzle and ensures that your hard work pays off with an accurate answer. Now that we've confirmed our solution, we can confidently say that we've mastered this equation!

Final Answer: m = 85

Alright, guys, we did it! We successfully solved the equation √m+15 - 1 = 9 and found that m = 85. We walked through each step, from isolating the square root to eliminating it and finally isolating 'm'. We even checked our solution to make sure it was correct. You've now got a solid understanding of how to tackle equations with square roots. Remember, the key is to use inverse operations, keep the equation balanced, and always check your work. Keep practicing, and you'll become a master equation solver in no time! Math can be challenging, but with a systematic approach and a little bit of patience, you can conquer any problem that comes your way. So, go forth and solve, my friends! And remember, if you ever get stuck, don't hesitate to break the problem down into smaller steps and tackle each one individually. You've got this!