Solution Of √{x²+49} = X + 5 A Comprehensive Guide
In mathematics, solving equations is a fundamental skill. This article delves into a specific algebraic problem: finding the solution to the equation √{x²+49} = x + 5. We will explore the steps involved in solving this equation, discuss potential pitfalls, and verify our solution to ensure accuracy. This comprehensive guide aims to provide a clear and thorough understanding of the solution process, suitable for students and math enthusiasts alike.
Understanding the Equation
Our main keyword in this discussion is the equation √{x²+49} = x + 5. This equation involves a square root, which adds a layer of complexity to the solution process. The presence of the square root necessitates careful consideration of potential extraneous solutions, which are solutions obtained algebraically but do not satisfy the original equation. Therefore, it's crucial to verify any solution we find by plugging it back into the original equation. Before diving into the algebraic manipulation, let's understand the components of this equation. The left-hand side features a square root of a quadratic expression (x² + 49), while the right-hand side is a linear expression (x + 5). Our goal is to find the value(s) of 'x' that make both sides of the equation equal. The general strategy for solving equations involving square roots is to isolate the square root term and then square both sides of the equation. This eliminates the square root, allowing us to solve for 'x' using standard algebraic techniques. However, squaring both sides can introduce extraneous solutions, hence the need for verification. Let's proceed with the solution step-by-step, emphasizing the importance of each step and the rationale behind it. The initial step is to ensure the square root is isolated, which in this case, it already is. Next, we'll square both sides to eliminate the radical and transform the equation into a more manageable form. After solving the resulting equation, we will meticulously check our solution(s) to ensure they are valid within the original equation's domain. Understanding these preliminary steps sets the stage for a smooth and accurate solution process, preventing common errors and ensuring a solid grasp of the underlying mathematical principles. This approach not only solves the problem at hand but also reinforces essential algebraic problem-solving skills.
Step-by-Step Solution
To solve the **equation √x²+49} = x + 5**, we'll follow a step-by-step approach. Firstly, squaring both sides of the equation is a key step to eliminate the square root. This gives us (√{x²+49})² = (x + 5)². Simplifying the left side, we get x² + 49, and expanding the right side, we obtain x² + 10x + 25. Now our equation looks like x² + 49 = x² + 10x + 25. Notice that we have x² terms on both sides. Subtracting x² from both sides simplifies the equation further, leaving us with 49 = 10x + 25. This is now a linear equation, which is much easier to solve. The next step is to isolate the 'x' term. We can do this by subtracting 25 from both sides of the equation = x + 5, to check its validity. This verification step is not just a formality; it's a necessary part of the solution process, ensuring that the value we found actually satisfies the initial conditions of the problem. Without this step, we risk accepting a solution that is algebraically correct but does not hold true in the context of the original equation, leading to an incorrect answer. By meticulously following these steps and emphasizing verification, we demonstrate a rigorous approach to solving equations involving square roots.
Verifying the Solution
Verifying our potential solution, x = 12/5, is a critical step. We substitute x = 12/5 back into the original equation, √{x²+49} = x + 5. This gives us √{(12/5)² + 49} = (12/5) + 5. Let's first evaluate the left-hand side. (12/5)² is 144/25. So, we have √{144/25 + 49}. To add these terms, we need a common denominator. We rewrite 49 as 49 * (25/25) = 1225/25. Therefore, the expression under the square root becomes 144/25 + 1225/25 = 1369/25. The square root of 1369/25 is √{1369} / √{25} = 37/5. Now, let's evaluate the right-hand side. (12/5) + 5 can be rewritten as (12/5) + (5 * 5 / 5) = 12/5 + 25/5 = 37/5. Comparing both sides, we have 37/5 = 37/5, which is a true statement. This confirms that x = 12/5 is indeed a valid solution to the original equation. This verification process highlights the importance of not just finding a potential solution but also ensuring its correctness by substituting it back into the original equation. It's a safeguard against extraneous solutions that may arise due to the squaring of both sides. A thorough verification process instills confidence in the final answer and demonstrates a complete understanding of the solution.
Potential Pitfalls and How to Avoid Them
When solving equations like √{x²+49} = x + 5, there are potential pitfalls to watch out for. One major pitfall is forgetting to verify the solution. Squaring both sides of an equation can introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. We avoided this by explicitly verifying that x = 12/5 is a valid solution. Another pitfall is making algebraic errors during the simplification process. For example, incorrectly expanding (x + 5)² or miscalculating the square root can lead to an incorrect solution. Careful and methodical algebraic manipulation is key to avoiding these errors. It's also important to remember the order of operations (PEMDAS/BODMAS) when simplifying expressions. Another potential issue arises from misunderstanding the domain of the square root function. The expression under the square root must be non-negative. However, in this specific case, x² + 49 is always positive for any real number x, so we don't have domain restrictions based on the square root. However, the right-hand side, x + 5, implicitly imposes a condition. For the square root to equal x + 5, x + 5 must be non-negative, i.e., x ≥ -5. Our solution, x = 12/5, satisfies this condition. Paying attention to such implicit conditions is vital for ensuring the solution's validity. To avoid these pitfalls, it's beneficial to double-check each step of the solution, be mindful of algebraic manipulations, and always verify the final solution. A systematic approach, coupled with careful attention to detail, minimizes the risk of making mistakes and ensures accurate results.
Conclusion
In conclusion, the solution to the equation √{x²+49} = x + 5 is x = 12/5. We arrived at this solution by squaring both sides of the equation, simplifying the resulting expression, and solving for x. We then rigorously verified our solution by substituting it back into the original equation, ensuring its validity. This process highlighted the importance of verification in solving equations involving square roots, as it helps to eliminate extraneous solutions. Throughout the solution, we emphasized the potential pitfalls, such as algebraic errors and neglecting to verify the solution, and provided strategies to avoid them. Understanding and addressing these pitfalls are crucial for developing robust problem-solving skills in mathematics. By following a systematic approach, paying attention to detail, and always verifying the solution, we can confidently solve such equations and avoid common mistakes. This comprehensive guide has demonstrated a step-by-step approach to solving this specific equation, while also imparting valuable insights into the general principles of solving algebraic equations involving square roots. The ability to solve such equations is a fundamental skill in mathematics, and a thorough understanding of the process is essential for further studies in algebra and related fields. The key takeaway is the importance of a systematic approach, careful algebraic manipulation, and rigorous verification to ensure accurate and reliable solutions.
