Solving The Equation 4 = Sqrt(m) - 8 And Verifying The Solution

Introduction

In this article, we will embark on a journey to solve the equation $4 = \sqrt{m} - 8$. This equation involves a square root, which adds a layer of complexity compared to simple linear equations. Our primary goal is to isolate the variable m and determine its value. Furthermore, a crucial aspect of solving equations, especially those involving radicals, is to check the solution we obtain. This verification step ensures that our solution is valid and satisfies the original equation. We will meticulously detail each step involved in solving for m and subsequently demonstrate how to verify our result. This process will not only provide a concrete solution to the given equation but also reinforce the fundamental principles of algebraic manipulation and solution verification.

Understanding the importance of checking the solution cannot be overstated, particularly when dealing with radical equations. Extraneous solutions, which are values that emerge during the solving process but do not satisfy the original equation, are a common occurrence. These extraneous solutions arise because the process of squaring both sides of an equation, a typical technique for eliminating square roots, can introduce solutions that were not present in the initial equation. Therefore, the act of substituting the calculated value back into the original equation acts as a safeguard, ensuring that we only accept solutions that genuinely make the equation true. This rigorous approach is essential for maintaining accuracy and avoiding errors in mathematical problem-solving.

Our exploration will begin with a step-by-step algebraic manipulation to isolate the square root term and eventually solve for m. Following this, we will delve into the verification process, showcasing how to substitute the calculated value back into the original equation to confirm its validity. This comprehensive approach aims to provide a clear and thorough understanding of how to effectively solve radical equations and verify the obtained solutions. This skill is not only vital in academic settings but also has practical applications in various fields that rely on mathematical modeling and problem-solving.

Step-by-Step Solution

To solve the equation $4 = \sqrt{m} - 8$, our initial objective is to isolate the square root term. This can be achieved by performing algebraic operations on both sides of the equation. We begin by adding 8 to both sides of the equation. This operation is based on the fundamental principle that adding the same value to both sides of an equation maintains the equality. By adding 8, we effectively eliminate the -8 term on the right side, bringing us closer to isolating the square root.

Adding 8 to both sides of the equation, we get:

$4 + 8 = \sqrt{m} - 8 + 8$

This simplifies to:

$12 = \sqrt{m}$

Now that we have isolated the square root term, the next step is to eliminate the square root. This is typically accomplished by squaring both sides of the equation. Squaring both sides is the inverse operation of taking the square root, effectively undoing the radical and allowing us to solve for m. However, it's important to remember that squaring both sides can sometimes introduce extraneous solutions, which we will address later in the verification step.

Squaring both sides of the equation $12 = \sqrt{m}$, we obtain:

$(12)^2 = (\sqrt{m})^2$

This simplifies to:

$144 = m$

Thus, we have arrived at a potential solution: $m = 144$. This value represents the solution we obtained through algebraic manipulation. However, before we definitively declare this as the solution, we must rigorously check the solution to ensure it satisfies the original equation and is not an extraneous solution. The verification process is crucial to the integrity of the solution, particularly in equations involving radicals.

In the subsequent section, we will detail the process of verifying this solution, demonstrating how to substitute the calculated value of m back into the original equation and confirm its validity. This step is an indispensable part of solving radical equations and guarantees the accuracy of our final answer.

Verifying the Solution

Now that we have obtained a potential solution, $m = 144$, it is imperative to check our solution to ensure it satisfies the original equation: $4 = \sqrt{m} - 8$. This verification step is particularly crucial when dealing with radical equations, as squaring both sides of an equation can sometimes introduce extraneous solutions, which are values that do not actually satisfy the initial equation.

To verify the solution, we will substitute $m = 144$ back into the original equation and evaluate both sides. If the left-hand side (LHS) equals the right-hand side (RHS) after the substitution, then our solution is valid. If the two sides are not equal, it indicates that $m = 144$ is an extraneous solution and not a valid solution to the equation.

Substituting $m = 144$ into the original equation, we get:

$4 = \sqrt{144} - 8$

Now we need to simplify the right-hand side of the equation. The square root of 144 is 12, since $12 * 12 = 144$. Therefore, the equation becomes:

$4 = 12 - 8$

Next, we perform the subtraction on the right-hand side:

$4 = 4$

We observe that the left-hand side (4) is equal to the right-hand side (4). This equality confirms that $m = 144$ is indeed a valid solution to the original equation. Since the substitution of $m = 144$ satisfies the equation, we can confidently conclude that it is the correct solution.

This solution verification process highlights the importance of not only finding a potential solution but also confirming its validity. It underscores the rigorous nature of mathematical problem-solving, where accuracy and precision are paramount. By meticulously checking our solution, we can be certain that our answer is correct and that we have successfully solved the equation.

In conclusion, after both solving the equation and verifying the solution, we have a complete and accurate understanding of the value of m that satisfies the given equation.

Conclusion

In this comprehensive exploration, we successfully solved the equation $4 = \sqrt{m} - 8$ and meticulously checked the solution to ensure its validity. Our journey began with isolating the square root term through algebraic manipulation, a process that involved adding 8 to both sides of the equation. This step is a cornerstone of equation solving, emphasizing the principle of maintaining balance while manipulating terms. By isolating the square root, we set the stage for the next crucial step: eliminating the radical.

The elimination of the square root was achieved by squaring both sides of the equation. This operation is the inverse of taking the square root and allows us to solve for m. However, it is essential to acknowledge that squaring both sides can potentially introduce extraneous solutions. This is a common phenomenon in radical equations and underscores the necessity of a rigorous verification process. The potential for extraneous solutions highlights the importance of not just finding a value for m but also confirming that this value truly satisfies the original equation.

Our algebraic manipulations led us to a potential solution: $m = 144$. However, we did not stop there. We proceeded to the crucial step of verifying this solution. This involved substituting $m = 144$ back into the original equation and evaluating both sides. The verification process is the gold standard for ensuring the accuracy of solutions in radical equations. It acts as a safeguard against extraneous solutions and confirms that the calculated value is indeed a valid solution.

By substituting $m = 144$ into the original equation, we found that both sides of the equation were equal, confirming that $m = 144$ is the correct solution. This verification step not only validated our solution but also reinforced the importance of this process in mathematical problem-solving. The act of verifying the solution is not merely a formality; it is an integral part of the problem-solving process, ensuring the accuracy and reliability of the final answer.

The process of solving and verifying this equation underscores the fundamental principles of algebra and the importance of meticulous problem-solving techniques. It showcases how algebraic manipulation, combined with a rigorous verification process, leads to accurate and reliable solutions. This approach is not only applicable to solving radical equations but also serves as a valuable framework for tackling various mathematical problems. The emphasis on checking the solution is a critical takeaway, highlighting the need for thoroughness and accuracy in mathematical endeavors.

In summary, we have demonstrated a step-by-step approach to solving the equation $4 = \sqrt{m} - 8$, emphasizing the importance of verifying the solution to ensure its validity. The final answer, $m = 144$, is not just a result of algebraic manipulation but a solution that has been rigorously tested and confirmed. This comprehensive approach underscores the core principles of mathematical problem-solving: accuracy, precision, and thoroughness.