Solving Square Root Equations: A Step-by-Step Guide To √x+2-15=-3

Hey guys! Let's dive into this intriguing mathematical problem together and unravel the mystery behind the equation $\sqrt{x+2}-15=-3$. This equation involves a square root, which might seem intimidating at first, but trust me, with a step-by-step approach, we can conquer it. Our ultimate goal is to isolate the variable 'x' and determine its value. So, buckle up, grab your thinking caps, and let's embark on this mathematical journey!

Step 1: Isolating the Square Root

The initial step in solving this equation is to isolate the square root term, which is $\sqrt{x+2}$. To achieve this, we need to get rid of the '-15' that's hanging around on the left side of the equation. Remember the golden rule of algebra: whatever you do to one side, you must do to the other! So, let's add 15 to both sides of the equation:

$\sqrt{x+2} - 15 + 15 = -3 + 15$

This simplifies to:

$\sqrt{x+2} = 12$

Great! We've successfully isolated the square root term. Now, the equation looks much cleaner and easier to handle. We're one step closer to finding the value of 'x'. This isolation is crucial because it allows us to deal directly with the square root and eliminate it in the next step. Imagine trying to solve for 'x' with the '-15' still there – it would be a messy situation! So, give yourselves a pat on the back for making it this far. You're doing fantastic!

The importance of isolating the square root cannot be overstated. It's like setting the stage for the main act – getting rid of the square root. Without this step, we'd be stuck in a maze of algebraic manipulations. Think of it as peeling away the layers of an onion, one step at a time, until we reach the core – the value of 'x'. And remember, in mathematics, just like in life, a clear and methodical approach is the key to success. So, keep practicing these steps, and you'll become a master equation solver in no time!

Step 2: Squaring Both Sides

Now that we've isolated the square root, the next logical step is to eliminate it altogether. How do we do that? By squaring! Remember that the square of a square root cancels out the radical, leaving us with just the expression inside the square root. Again, we must adhere to the golden rule – whatever we do to one side, we must do to the other. So, let's square both sides of the equation:

$(\sqrt{x+2})^2 = 12^2$

This simplifies to:

$x + 2 = 144$

Voila! The square root is gone, and we're left with a simple linear equation. This is a huge victory! We've transformed a seemingly complex equation into a much more manageable form. The squaring operation is the magic trick that unlocks the door to solving for 'x'. It's like using a special key to open a locked chest, revealing the treasure inside – the solution to our equation.

Squaring both sides is a fundamental technique when dealing with square roots in equations. It's a powerful tool that allows us to get rid of the radical and work with a more familiar algebraic expression. However, it's also important to be cautious when squaring both sides, as this operation can sometimes introduce extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation. We'll need to check our final answer later to make sure it's a valid solution.

Step 3: Solving for x

We're in the home stretch now! We have the equation $x + 2 = 144$. This is a simple linear equation that we can solve by isolating 'x'. To do this, we need to get rid of the '+2' on the left side. You guessed it – we'll subtract 2 from both sides of the equation:

$x + 2 - 2 = 144 - 2$

This simplifies to:

$x = 142$

Eureka! We've found a potential solution for 'x'. But before we jump for joy, we need to remember the cautionary note from the previous step about extraneous solutions. Squaring both sides can sometimes lead to solutions that don't actually work in the original equation. So, we must verify our solution.

Isolating 'x' is the ultimate goal in solving any equation. It's like the final piece of the puzzle that completes the picture. In this case, we achieved isolation by subtracting 2 from both sides, effectively undoing the addition operation. This is a common technique in algebra, and mastering it is crucial for solving a wide range of equations. Think of it as a balancing act – we're carefully manipulating the equation while keeping both sides equal. And remember, every step we take brings us closer to the ultimate answer.

Step 4: Verifying the Solution

Okay, we've got a potential solution: $x = 142$. But before we declare victory, we need to verify this solution. This is a crucial step, especially when dealing with equations involving square roots, as squaring both sides can sometimes introduce extraneous solutions. To verify, we'll plug $x = 142$ back into the original equation:

$\sqrt{x+2}-15=-3$

Substituting $x = 142$, we get:

$\sqrt{142+2}-15=-3$

Simplifying the expression inside the square root:

$\sqrt{144}-15=-3$

The square root of 144 is 12, so:

$12 - 15 = -3$

And finally:

$-3 = -3$

It checks out! Our solution, $x = 142$, satisfies the original equation. This means it's a valid solution, and we can confidently say that we've solved the equation.

Verification is the final safeguard in the problem-solving process. It's like the quality control check that ensures our answer is accurate and reliable. In this case, verification involved substituting our potential solution back into the original equation and confirming that it holds true. This step is particularly important when dealing with operations like squaring both sides, which can sometimes lead to extraneous solutions. Think of verification as the ultimate test – if our solution passes, we know we've done everything correctly. So, never skip this step, and you'll avoid falling into the trap of extraneous solutions.

The Final Answer

After meticulously working through each step, we've arrived at the final answer. We isolated the square root, squared both sides, solved for 'x', and, most importantly, verified our solution. The solution to the equation $\sqrt{x+2}-15=-3$ is:

$x = 142$

So, the correct answer is A. $x=142$.

Congratulations! You've successfully navigated the world of square root equations and emerged victorious. Remember, the key to solving these types of problems is a systematic approach: isolate, eliminate, solve, and verify. Keep practicing, and you'll become a master of mathematical problem-solving. And never forget the importance of verification – it's the final step that ensures your answer is correct. So, go forth and conquer more mathematical challenges!