Solving √{x+2} - 15 = -3 Step-by-Step Guide

Introduction

In mathematics, solving equations is a fundamental skill. This article provides a step-by-step guide on how to solve the equation √{x+2} - 15 = -3. We will explore the process of isolating the variable x and verifying the solution. Understanding these steps is crucial for tackling more complex algebraic problems. This comprehensive guide aims to equip you with the knowledge and confidence to solve similar equations effectively. Solving equations often involves isolating the variable, and this equation is no different. We'll focus on each step, ensuring clarity and understanding for readers of all mathematical backgrounds. The initial equation, √{x+2} - 15 = -3, might seem daunting, but breaking it down into manageable steps makes the process straightforward. Let's embark on this mathematical journey together, unraveling the solution to this equation and solidifying your understanding of algebraic problem-solving.

Step 1: Isolate the Square Root

The first crucial step in solving the equation √x+2} - 15 = -3 is to isolate the square root term. This means we need to get the term √{x+2} by itself on one side of the equation. To do this, we will add 15 to both sides of the equation. This maintains the balance of the equation and moves us closer to isolating the square root. Adding 15 to both sides gives us √{x+2 - 15 + 15 = -3 + 15. Simplifying this, we get √{x+2} = 12. Now, the square root term is isolated, and we can proceed to the next step. Isolating the square root is a common technique when dealing with equations involving radicals. This prepares the equation for the next step, which involves eliminating the square root. By adding 15 to both sides, we've effectively simplified the equation and made it easier to work with. This is a crucial foundation for the subsequent steps in solving for x. This step underscores the importance of maintaining balance in equations, a fundamental principle in algebra.

Step 2: Eliminate the Square Root

Now that we have isolated the square root term, √{x+2} = 12, the next step is to eliminate the square root. To do this, we will square both sides of the equation. Squaring both sides is the inverse operation of taking the square root and will allow us to remove the radical. When we square both sides, we get (√{x+2})^2 = 12^2. This simplifies to x + 2 = 144. By squaring both sides, we have successfully eliminated the square root and transformed the equation into a simpler linear equation. Eliminating the square root is a pivotal step in solving radical equations. It allows us to work with a more straightforward equation, making it easier to isolate the variable x. This process highlights the power of inverse operations in mathematics. Squaring both sides is a valid operation as long as we remember to check our solutions later to ensure they are not extraneous. This step brings us closer to finding the value of x.

Step 3: Solve for x

After eliminating the square root, we have the equation x + 2 = 144. To solve for x, we need to isolate x on one side of the equation. This can be achieved by subtracting 2 from both sides of the equation. Subtracting 2 from both sides, we get x + 2 - 2 = 144 - 2. This simplifies to x = 142. Therefore, our potential solution is x = 142. Solving for x in this step involves a simple algebraic manipulation. By subtracting 2 from both sides, we've successfully isolated x and found a potential solution. However, it's crucial to remember that we need to verify this solution in the original equation to ensure it's not extraneous. This step demonstrates the importance of inverse operations in solving equations. Subtracting 2 is the inverse operation of adding 2, allowing us to isolate x. We're now one step closer to the final answer, but verification is key.

Step 4: Verify the Solution

It is essential to verify the solution we found in the original equation to ensure it is not extraneous. This is particularly important when dealing with radical equations because squaring both sides can sometimes introduce solutions that do not satisfy the original equation. We found that x = 142, so we will substitute this value back into the original equation: √{x+2} - 15 = -3. Substituting x = 142, we get √{142+2} - 15 = -3. Simplifying, we have √{144} - 15 = -3. The square root of 144 is 12, so we have 12 - 15 = -3. This simplifies to -3 = -3, which is a true statement. Since the equation holds true when x = 142, we can confirm that this is a valid solution. Verifying the solution is a critical step in solving any equation, especially those involving radicals. It ensures that the solution we found is indeed a valid solution and not an extraneous one. This step underscores the importance of rigor in mathematical problem-solving. By substituting the solution back into the original equation, we can confirm its validity. This step completes the process of solving the equation and gives us confidence in our answer.

Conclusion

In conclusion, the solution to the equation √{x+2} - 15 = -3 is x = 142. We arrived at this solution by following a step-by-step approach, which included isolating the square root, eliminating the square root by squaring both sides, solving for x, and verifying the solution in the original equation. This process demonstrates the importance of understanding algebraic principles and applying them systematically. Solving equations is a fundamental skill in mathematics, and mastering this skill opens doors to more advanced mathematical concepts. By understanding the steps involved in solving radical equations, you can confidently tackle similar problems. The key takeaways from this guide include the importance of isolating the radical, using inverse operations, and verifying solutions. This comprehensive approach ensures accuracy and a deeper understanding of the underlying mathematical principles. We hope this guide has been helpful in clarifying the process of solving radical equations.