Solving The Equation √a + 11 = 21 A Step By Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving the equation √a + 11 = 21. We will explore the steps involved in isolating the variable 'a' and finding its value. This comprehensive guide aims to provide a clear understanding of the process, making it accessible to students and enthusiasts alike. Understanding radical equations is crucial, and this article serves as an excellent resource for mastering such problems. Our main focus will be on dissecting the equation, applying algebraic principles, and arriving at the solution in a step-by-step manner. This journey through solving √a + 11 = 21 will not only enhance your problem-solving skills but also deepen your understanding of mathematical concepts. We'll begin by isolating the radical term, then squaring both sides, and finally, verifying the solution to ensure accuracy. This methodical approach is key to tackling any algebraic problem, and this article will guide you through each stage with clarity and precision. Furthermore, we'll emphasize the importance of checking for extraneous solutions, a common pitfall in solving radical equations. By the end of this guide, you'll be well-equipped to handle similar equations and confident in your mathematical abilities.
Step-by-Step Solution
Isolating the Radical Term
The initial step in solving the equation √a + 11 = 21 involves isolating the radical term, which in this case is √a. This is achieved by subtracting 11 from both sides of the equation. Isolating the square root is a critical first step. This process ensures that we have the radical term alone on one side, making it easier to eliminate the square root in the subsequent steps. When we subtract 11 from both sides, the equation transforms as follows:
√a + 11 - 11 = 21 - 11
This simplifies to:
√a = 10
Now that the radical term √a is isolated, we can proceed to the next step, which involves squaring both sides of the equation. Successfully isolating the radical is a testament to understanding basic algebraic manipulations. This sets the stage for eliminating the radical and solving for 'a'. The ability to perform this isolation accurately is fundamental to solving radical equations. Therefore, mastering this step is crucial for anyone seeking to excel in algebra and related mathematical fields. We'll continue to build upon this foundation as we move forward in the solution process, ensuring that each step is clear, concise, and easy to follow.
Squaring Both Sides
With the radical term isolated, the next crucial step is to eliminate the square root. We achieve this by squaring both sides of the equation √a = 10. Squaring both sides is a fundamental operation in solving equations involving radicals. Squaring both sides of √a = 10 allows us to remove the square root and obtain a simpler equation. This process relies on the principle that if two quantities are equal, then their squares are also equal. Applying this principle, we square both √a and 10:
(√a)² = 10²
This simplifies to:
a = 100
By squaring both sides, we have successfully eliminated the square root and obtained a value for 'a'. However, it's essential to remember that squaring both sides can sometimes introduce extraneous solutions. Therefore, the next step involves verifying this solution to ensure its validity. The act of eliminating the square root is a significant milestone in solving the equation. It transforms the equation into a more manageable form, allowing us to directly solve for the variable 'a'. The result, a = 100, is a potential solution, but we must proceed with caution and verify it before declaring it as the final answer. This meticulous approach ensures the accuracy of our solution and reinforces the importance of thoroughness in mathematical problem-solving.
Verifying the Solution
After obtaining a potential solution, it's imperative to verify it in the original equation. This step is crucial to ensure that the solution is valid and not an extraneous solution introduced during the squaring process. Verifying the solution a = 100 is a critical step in solving radical equations. Extraneous solutions can arise when squaring both sides of an equation, so verification is essential to ensure accuracy. To verify the solution a = 100, we substitute it back into the original equation:
√a + 11 = 21
Substituting a = 100, we get:
√100 + 11 = 21
Simplifying the square root, we have:
10 + 11 = 21
This further simplifies to:
21 = 21
Since the equation holds true, the solution a = 100 is valid. This verification step confirms that our solution is correct and not an extraneous solution. The confirmation that 21 = 21 validates our solution. It demonstrates that the value we obtained for 'a' satisfies the original equation, providing us with confidence in our answer. This process of verification is not merely a formality; it is an integral part of the problem-solving process, ensuring the correctness and reliability of the solution. Therefore, always remember to verify your solutions when dealing with radical equations to avoid the pitfall of extraneous solutions and to ensure the accuracy of your mathematical work.
Conclusion
In conclusion, the solution to the equation √a + 11 = 21 is a = 100. We arrived at this solution by systematically isolating the radical term, squaring both sides of the equation, and, most importantly, verifying the solution in the original equation. The final solution, a = 100, represents the culmination of our step-by-step approach. This solution has been rigorously verified, ensuring its accuracy and validity. Throughout this process, we have emphasized the importance of each step, from isolating the radical to checking for extraneous solutions. The ability to solve the equation √a + 11 = 21 demonstrates a strong understanding of algebraic principles and problem-solving techniques. This methodical approach is applicable to a wide range of mathematical problems, making it a valuable skill to develop. Furthermore, the process of verifying the solution highlights the importance of precision and thoroughness in mathematics. By adhering to these principles, we can confidently tackle complex equations and arrive at accurate solutions. This comprehensive guide has aimed to provide clarity and understanding, empowering you to approach similar problems with confidence and expertise. Remember, practice and attention to detail are key to mastering mathematical concepts, and this solution serves as a solid foundation for further exploration in the world of algebra.
