Solving The Equation √k+7 = 3√2 A Step-by-Step Guide
In this article, we delve into the process of solving the equation √k+7 = 3√2, a mathematical problem that falls under the category of algebra and radical equations. Our goal is to provide a step-by-step, comprehensive guide that not only helps you understand the solution but also equips you with the knowledge to tackle similar problems. This discussion will cover the fundamental principles of algebraic manipulation, the properties of square roots, and the techniques used to isolate variables and solve equations. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, this article aims to offer clarity and practical insights into solving equations involving square roots. The journey through this equation will highlight the importance of understanding the underlying mathematical concepts and the strategic application of algebraic rules to arrive at a solution. By the end of this guide, you should feel confident in your ability to solve equations of this nature and appreciate the elegance of mathematical problem-solving. We'll start by dissecting the equation, identifying the key components, and then move on to the solution process, explaining each step in detail.
Understanding the Equation: √k+7 = 3√2
The equation √k+7 = 3√2 presents a classic algebraic challenge involving square roots. To begin, let's break down the equation and identify its components. On the left side, we have the square root of the expression 'k+7', where 'k' is the variable we aim to solve for. The presence of the square root indicates that we'll need to employ specific techniques to isolate 'k', which will likely involve squaring both sides of the equation at some point. The '7' within the square root adds a layer of complexity, as it's directly connected to 'k' under the radical. On the right side, we have '3√2', which is a constant value. Here, '√2' represents the square root of 2, and it's multiplied by 3. This term gives us a numerical benchmark that the left side of the equation must equal. Solving this equation requires us to carefully apply the rules of algebra, particularly those related to square roots and exponents. The process will involve isolating the square root term, squaring both sides to eliminate the radical, and then further algebraic manipulation to finally solve for 'k'. Understanding the structure of the equation and the nature of each term is crucial for developing an effective strategy to find the solution. The initial assessment allows us to anticipate the steps involved and appreciate the logical flow required to unravel the equation. As we progress, we'll see how each step builds upon the previous one, leading us closer to the final answer.
Step-by-Step Solution
Isolating the Square Root
The first crucial step in solving the equation √k+7 = 3√2 is to isolate the square root term. In this case, the square root term is already isolated on the left side of the equation, which simplifies our initial task. The term √k+7 is the focal point, and our next action will directly target this radical. Since the square root is isolated, we can proceed to the next logical step, which involves eliminating the square root itself. This isolation is a key principle in solving radical equations, as it allows us to apply the inverse operation—squaring—to both sides of the equation. Without this initial isolation, we would encounter difficulties in properly simplifying the equation and could potentially introduce errors in our calculations. Therefore, recognizing that the square root is already isolated is a significant advantage in this particular problem. It allows us to move forward efficiently and focus on the subsequent algebraic manipulations needed to solve for 'k'. This emphasis on isolation highlights the importance of carefully observing the structure of the equation before diving into the solution process. It's a reminder that a strategic approach, beginning with isolating the radical, can often lead to a more straightforward and successful resolution. As we move forward, we'll see how this initial step sets the stage for the rest of the solution.
Squaring Both Sides
To eliminate the square root in the equation √k+7 = 3√2, we perform the operation of squaring both sides. This is a fundamental algebraic technique used to deal with radical equations. Squaring both sides ensures that we maintain the equality of the equation while getting rid of the square root. When we square the left side, (√k+7)², the square root and the square cancel each other out, leaving us with simply 'k+7'. On the right side, we square '3√2', which means we calculate (3√2)². This can be broken down into 3² * (√2)². 3² equals 9, and (√2)² equals 2, so the right side becomes 9 * 2, which equals 18. So, after squaring both sides, our equation transforms from √k+7 = 3√2 to k+7 = 18. This transformation is a critical step in solving for 'k', as it removes the square root, making the equation much easier to manipulate algebraically. The key here is to remember that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain balance. Squaring both sides is a direct application of this principle, and it's a common technique used in solving a wide variety of algebraic equations, especially those involving radicals. With the square root now eliminated, we have a simple linear equation that we can easily solve in the next step.
Solving for 'k'
After squaring both sides, our equation is now simplified to k+7 = 18. To solve for 'k', we need to isolate 'k' on one side of the equation. This can be achieved by subtracting 7 from both sides. When we subtract 7 from the left side, k+7 - 7, the 7s cancel each other out, leaving us with just 'k'. On the right side, we subtract 7 from 18, which gives us 18 - 7 = 11. Therefore, the equation becomes k = 11. This is our solution for 'k'. We have successfully isolated 'k' and found its value. The process of subtracting 7 from both sides is another example of maintaining balance in an equation. It's a basic algebraic principle that ensures the equality remains valid throughout the solution process. With 'k' now isolated and its value determined, we have completed the algebraic manipulation necessary to solve the equation. Our final step will be to verify this solution to ensure its accuracy. Solving for 'k' in this manner demonstrates the power of algebraic manipulation and the importance of following a logical sequence of steps to arrive at the solution. The transition from the radical equation to this simple linear form showcases the effectiveness of squaring both sides as a technique for eliminating square roots.
Verification
To ensure the accuracy of our solution, we must verify that k = 11 indeed satisfies the original equation, √k+7 = 3√2. Verification is a crucial step in solving any equation, especially those involving radicals, as it helps us catch any potential errors made during the algebraic manipulation. To verify, we substitute the value of k, which is 11, back into the original equation. This gives us √11+7 = 3√2. Now, we simplify the left side of the equation. 11 + 7 equals 18, so we have √18. The square root of 18 can be simplified further. We recognize that 18 is equal to 9 multiplied by 2, so √18 can be written as √(9 * 2). Using the property of square roots, we can separate this into √9 * √2. The square root of 9 is 3, so we now have 3√2. Comparing this to the right side of the original equation, which is also 3√2, we see that both sides are equal. This confirms that our solution, k = 11, is correct. The verification process not only validates our solution but also reinforces our understanding of the steps taken. It provides a sense of closure and confidence in the answer we've obtained. The ability to verify a solution is a valuable skill in mathematics, as it empowers us to check our work and ensure accuracy.
Alternative Approaches
While the step-by-step solution provides a direct method for solving the equation √k+7 = 3√2, exploring alternative approaches can deepen our understanding and offer different perspectives on problem-solving. One alternative approach involves simplifying the right side of the equation before squaring. We have 3√2 on the right side, which is already in its simplest radical form. However, recognizing this explicitly can be a helpful starting point. Another perspective is to consider the domain of the variable 'k'. Since we have a square root, the expression inside the square root, 'k+7', must be non-negative. This means k+7 ≥ 0, which implies k ≥ -7. This initial consideration of the domain can be crucial in more complex problems where extraneous solutions might arise. In this particular case, our solution k=11 satisfies this condition. Furthermore, one could also approach the problem graphically. While not a direct algebraic method, visualizing the equation as two functions, y = √k+7 and y = 3√2, and finding their intersection point can provide a visual confirmation of the solution. The x-coordinate of the intersection point would represent the value of 'k' that satisfies the equation. Exploring these alternative approaches not only enhances our problem-solving skills but also demonstrates the interconnectedness of different mathematical concepts. It reinforces the idea that there isn't always one single path to a solution and that flexibility in thinking can be a valuable asset in mathematical problem-solving. Each approach offers a unique lens through which to view the problem, contributing to a more comprehensive understanding.
Conclusion
In conclusion, we have successfully navigated the solution of the equation √k+7 = 3√2 through a detailed, step-by-step process. Starting with an understanding of the equation's components, we isolated the square root term, squared both sides to eliminate the radical, solved for 'k', and then verified our solution. This journey through the equation highlights several key principles of algebraic manipulation and problem-solving. We emphasized the importance of isolating the radical term before squaring, a crucial step in handling radical equations. The technique of squaring both sides was instrumental in removing the square root, allowing us to transform the equation into a simpler linear form. The process of solving for 'k' involved basic algebraic operations, such as subtracting the same value from both sides, to maintain the equation's balance. Verification served as a critical safeguard, ensuring the accuracy of our solution and reinforcing our understanding of the steps taken. Additionally, we explored alternative approaches, demonstrating the flexibility and interconnectedness of mathematical concepts. These alternative perspectives offer valuable insights and broaden our problem-solving toolkit. By mastering the techniques presented in this guide, you can confidently approach similar equations and develop a deeper appreciation for the elegance and power of algebra. The ability to solve equations like this is not only a valuable mathematical skill but also a testament to the power of logical thinking and systematic problem-solving. This comprehensive exploration aims to equip you with the knowledge and confidence to tackle mathematical challenges with greater ease and understanding.
