Solving √(c+10) = 4 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Solving equations involves finding the values of the unknown variables that satisfy a given equality. Among various types of equations, those involving square roots often pose a unique challenge. This article delves into the step-by-step solution of the equation ${\sqrt{c+10}=4}$, providing a comprehensive explanation of each stage. This exploration is designed to enhance your understanding of algebraic manipulations, especially those related to square roots. We will begin by outlining the initial equation and the goal of isolating the variable 'c'. Subsequently, we will detail the process of squaring both sides to eliminate the square root, followed by the necessary algebraic steps to solve for 'c'. Furthermore, we will emphasize the critical importance of verifying the solution to ensure it is not extraneous. This detailed approach not only solves the specific equation but also equips you with the knowledge to tackle similar problems with confidence. Understanding the nuances of square root equations is essential for various mathematical contexts, including algebra, calculus, and beyond. Thus, mastering these techniques is a valuable asset in your mathematical journey. Remember, the key to solving any equation lies in systematically applying the rules of algebra and ensuring that each step is logically sound. By carefully following the steps outlined in this article, you will gain a solid understanding of how to solve square root equations effectively. Let's embark on this mathematical journey together!

Understanding the Equation

The equation we aim to solve is ${\sqrt{c+10}=4}$. At its core, this equation states that the square root of the expression 'c+10' is equal to 4. To decipher this, we must isolate 'c' – the variable we are trying to find. The presence of the square root necessitates a specific approach, as we cannot directly manipulate 'c' while it is under the radical. Isolating the variable is the primary objective in solving any algebraic equation, and in this case, it requires us to first eliminate the square root. This is typically achieved by performing the inverse operation, which, in this scenario, is squaring. Squaring both sides of the equation will effectively undo the square root, allowing us to work with a simpler, linear equation. However, it's crucial to remember that squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, verification of the final solution is a vital step in the process. Before we proceed with squaring, it's worth noting the domain of the variable. Since we are dealing with a square root, the expression inside the root (c+10) must be non-negative. This means that 'c+10' must be greater than or equal to zero, which implies that 'c' must be greater than or equal to -10. This constraint is important to keep in mind as we solve the equation and verify our solution. The goal here is not just to find a value for 'c' but to find a value that makes the original equation true and adheres to the domain restriction. With a clear understanding of the equation and the steps involved, we can now proceed to the solution process.

Step-by-Step Solution

The initial step in solving the equation ${\sqrt{c+10}=4}$ is to eliminate the square root. To achieve this, we square both sides of the equation. Squaring both sides is a valid algebraic manipulation as long as we apply it to the entire equation, maintaining the balance. When we square the left side, ${\sqrt{c+10}}$, the square root is effectively canceled out, leaving us with 'c+10'. On the right side, squaring 4 results in 16. Therefore, after squaring both sides, the equation transforms into: ${c + 10 = 16}$. This new equation is a simple linear equation, which is much easier to solve than the original square root equation. Now, to isolate 'c', we need to subtract 10 from both sides of the equation. This is another fundamental algebraic operation that preserves the equality. Subtracting 10 from both sides gives us: ${c = 16 - 10}$. Performing the subtraction, we find that: ${c = 6}$. So, we have arrived at a potential solution: 'c' equals 6. However, as mentioned earlier, it is crucial to verify this solution to ensure it is not extraneous. Extraneous solutions can arise when squaring both sides of an equation, as this operation can sometimes introduce values that satisfy the squared equation but not the original equation. Therefore, we must substitute 'c = 6' back into the original equation to confirm its validity. This verification step is a cornerstone of solving square root equations and should never be overlooked. The next section will detail the verification process, ensuring that our solution is indeed correct.

Verifying the Solution

After finding a potential solution, the critical step of verification follows. In the context of square root equations, verifying the solution is not merely a formality; it's an essential safeguard against extraneous solutions. To verify our solution, ${c = 6}$, we substitute this value back into the original equation: ${\sqrt{c+10}=4}$. Substituting ${c = 6}$ gives us: ${\sqrt{6+10}=4}$. Simplifying the expression inside the square root, we get: ${\sqrt{16}=4}$. Now, we evaluate the square root of 16, which is 4. Thus, the equation becomes: ${4 = 4}$. This equality holds true, which confirms that ${c = 6}$ is indeed a valid solution to the original equation. The verification process not only confirms the correctness of the solution but also reinforces the understanding of the equation's behavior. It highlights the importance of ensuring that the solution satisfies the original conditions set forth by the equation. In this case, the square root of 'c+10' must equal 4, and our solution, ${c = 6}$, meets this criterion. Furthermore, the verification step implicitly checks whether the solution falls within the domain of the variable. Recall that we mentioned earlier that 'c' must be greater than or equal to -10. Our solution, ${c = 6}$, clearly satisfies this condition. This comprehensive verification process provides us with confidence that ${c = 6}$ is the correct and complete solution to the equation ${\sqrt{c+10}=4}$. With the solution verified, we can now confidently state the final answer and reflect on the process we undertook to arrive at it.

Final Answer

Having meticulously solved the equation ${\sqrt{c+10}=4}$ and rigorously verified the solution, we can now confidently state the final answer. The solution to the equation is: ${c = 6}$. This result signifies the value of 'c' that satisfies the given equation. The final answer is the culmination of a series of logical steps, each carefully executed to ensure accuracy. We began by understanding the equation and the goal of isolating the variable 'c'. We then employed the algebraic technique of squaring both sides to eliminate the square root, transforming the equation into a simpler linear form. Subsequently, we solved for 'c' by performing basic arithmetic operations. Crucially, we emphasized the importance of verifying the solution by substituting it back into the original equation. This verification step confirmed that our solution was not extraneous and that it indeed satisfied the original equation. The process of solving this equation underscores the fundamental principles of algebra and the importance of systematic problem-solving. It also highlights the necessity of paying attention to detail and verifying solutions, especially when dealing with equations involving square roots or other non-linear functions. The solution ${c = 6}$ represents not just a numerical answer but a testament to the power of algebraic manipulation and logical reasoning. It is the result of a careful and methodical approach, demonstrating the effectiveness of mathematical techniques in solving complex problems. In conclusion, the solution to the equation ${\sqrt{c+10}=4}$ is ${c = 6}$, a result we have arrived at with confidence and clarity.

Common Mistakes and How to Avoid Them

Solving equations, especially those involving square roots, can be prone to certain errors if not approached with caution. Recognizing common mistakes is crucial for developing accurate problem-solving skills. One of the most frequent errors is neglecting to verify the solution. As we've emphasized throughout this article, squaring both sides of an equation can introduce extraneous solutions. Failing to check the solution in the original equation can lead to accepting an incorrect answer. To avoid this, always substitute the solution back into the original equation and ensure that the equality holds true. Another common mistake is incorrectly applying the squaring operation. It's essential to square both sides of the equation in their entirety. For instance, if there are multiple terms on one side, squaring should apply to the entire expression, not individual terms. Misunderstanding the domain of the variable is another potential pitfall. Square roots are only defined for non-negative values. Therefore, the expression inside the square root must be greater than or equal to zero. Failing to consider this constraint can lead to solutions that are mathematically invalid. To avoid this, always consider the domain restrictions imposed by the square root. Arithmetic errors, while seemingly trivial, can also derail the solution process. Simple mistakes in addition, subtraction, multiplication, or division can propagate through the steps and result in an incorrect final answer. To minimize these errors, it's helpful to double-check each arithmetic operation and work in a clear and organized manner. Lastly, rushing through the solution process can lead to overlooking critical details. Taking the time to carefully analyze the equation, plan the steps, and execute them methodically can significantly reduce the likelihood of errors. By being aware of these common mistakes and actively working to avoid them, you can enhance your accuracy and confidence in solving square root equations. Remember, patience and attention to detail are key to success in mathematics.

Practice Problems

To solidify your understanding of solving square root equations, engaging with practice problems is invaluable. The more you practice, the more comfortable and proficient you will become with the techniques involved. Here are a few practice problems that are similar to the one we solved in this article:

1. Solve for x: ${\sqrt{x+5}=3}$
2. Solve for y: ${\sqrt{2y-1}=5}$
3. Solve for z: ${\sqrt{3z+4}=2}$
4. Solve for a: ${\sqrt{a-7}=1}$
5. Solve for b: ${\sqrt{4b+9}=7}$

These problems offer a range of scenarios that will challenge your understanding of the steps involved in solving square root equations. As you work through these problems, remember to follow the same methodical approach we outlined in this article. First, isolate the square root term. Then, square both sides of the equation to eliminate the square root. Next, solve the resulting linear equation for the variable. Finally, and most importantly, verify your solution by substituting it back into the original equation. This verification step will ensure that your solution is not extraneous. In addition to these problems, you can also create your own variations by changing the constants and coefficients. This will help you develop a deeper understanding of the underlying principles and improve your problem-solving skills. Remember, the key to mastering any mathematical concept is consistent practice. The more you work with different types of problems, the more adept you will become at recognizing patterns, applying the appropriate techniques, and arriving at accurate solutions. So, take the time to work through these practice problems, and you will undoubtedly see improvement in your ability to solve square root equations.

Conclusion

In conclusion, solving the equation ${\sqrt{c+10}=4}$ provides a valuable lesson in algebraic problem-solving. In conclusion, this exploration has taken us through the essential steps of dealing with square root equations, from understanding the initial equation to verifying the final solution. We began by recognizing the need to isolate the variable 'c' and the challenge posed by the square root. We then employed the technique of squaring both sides to eliminate the square root, transforming the equation into a more manageable linear form. Solving for 'c', we arrived at a potential solution, but we emphasized the crucial importance of verification. Substituting the solution back into the original equation, we confirmed its validity and ensured that it was not extraneous. This verification step is a cornerstone of solving square root equations and cannot be overemphasized. Throughout the process, we highlighted common mistakes to avoid, such as neglecting verification, misapplying the squaring operation, and overlooking domain restrictions. By being aware of these potential pitfalls, you can enhance your accuracy and confidence in solving similar problems. The practice problems provided offer an opportunity to further solidify your understanding and develop your skills. Consistent practice is key to mastering any mathematical concept, and square root equations are no exception. The solution to the equation, ${c = 6}$, represents not just a numerical answer but a demonstration of the power of algebraic manipulation and logical reasoning. It is the result of a systematic approach, careful execution, and diligent verification. By mastering the techniques outlined in this article, you will be well-equipped to tackle a wide range of algebraic problems, building a strong foundation for further mathematical studies. Remember, the journey of learning mathematics is a process of continuous exploration and refinement. Embrace the challenges, learn from your mistakes, and celebrate your successes. With perseverance and dedication, you can unlock the beauty and power of mathematics.