Solving -6√1x = 12 And Determining The Missing Value

Introduction

In this article, we will delve into the mathematical equation -6√1x = 12 and meticulously explore the steps required to solve it. Our primary goal is to determine the value of 'x' that satisfies this equation. Furthermore, we will address the question of what value needs to be added to the solution to reach 12. This exploration will not only enhance our understanding of algebraic equations but also provide valuable insights into problem-solving strategies in mathematics. We will break down each step in detail, ensuring clarity and comprehension for readers of all backgrounds. By the end of this article, you will have a comprehensive understanding of how to solve this particular equation and similar mathematical problems. The ability to manipulate and solve equations is a fundamental skill in mathematics, and this article aims to strengthen this skill through a clear and concise explanation.

Understanding the Equation

Before diving into the solution, it’s crucial to understand the components of the equation -6√1x = 12. This equation involves several key elements: a square root, a variable, and coefficients. The square root symbol (√) indicates that we are looking for a number that, when multiplied by itself, equals the expression under the root. In this case, we have √1x, which simplifies to √x since 1 multiplied by x is simply x. The coefficient -6 is multiplied by the square root of x, and the entire expression is set equal to 12. Understanding these components is the first step in solving the equation. It allows us to approach the problem methodically, applying the correct algebraic operations to isolate the variable and find its value. We will explore each of these components in more detail as we progress through the solution, ensuring a solid foundation for understanding the more complex steps.

Step 1: Isolate the Square Root

The first step in solving the equation -6√x = 12 is to isolate the square root term. This means we need to get √x by itself on one side of the equation. To do this, we will divide both sides of the equation by -6. This operation cancels out the -6 on the left side, leaving us with the square root term. Dividing both sides by the same number maintains the equality of the equation, which is a fundamental principle in algebra. This step is crucial because it simplifies the equation, making it easier to solve for x. By isolating the square root, we set the stage for the next step, which involves eliminating the square root altogether. This methodical approach ensures that we are applying the correct operations in the correct order, leading to an accurate solution. The importance of isolating the variable or term containing the variable cannot be overstated in solving algebraic equations.

Step 2: Eliminate the Square Root

After isolating the square root in the equation, we have √x = -2. The next step is to eliminate the square root. To do this, we square both sides of the equation. Squaring a square root cancels out the radical, leaving us with the variable x. However, it is essential to remember that squaring both sides of an equation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, it is crucial to check our solution in the original equation to ensure it is valid. Squaring both sides is a common technique used to solve equations involving radicals, and it is a critical step in finding the value of x. This step transforms the equation into a simpler form that we can easily solve.

Step 3: Solving for x

After squaring both sides of the equation √x = -2, we obtain x = 4. This is the value of x that we have found by solving the equation. However, we must now verify whether this solution is valid by substituting it back into the original equation. This step is crucial to ensure that we have not introduced any extraneous solutions during the process of squaring both sides. If the solution does not satisfy the original equation, it is not a valid solution. This verification process is a fundamental part of solving equations, especially those involving radicals. By checking our solution, we can be confident that we have found the correct value of x.

Step 4: Checking the Solution

To check our solution, we substitute x = 4 back into the original equation -6√x = 12. This gives us -6√4 = 12, which simplifies to -6 * 2 = 12, and further simplifies to -12 = 12. This statement is false, which means that x = 4 is not a valid solution to the original equation. The reason for this is that the square root of a number is always non-negative, and therefore, √x cannot be -2. This highlights the importance of checking solutions, especially when dealing with square roots and other radicals. In this case, the equation has no real solution because the square root of a number cannot be negative. This understanding is crucial for solving more complex equations in the future.

Determining the Missing Value to Reach 12

Now that we have established that the equation -6√1x = 12 has no real solution, we can move on to the second part of the question: how much is needed to reach 12? Since the left side of the equation simplifies to -12 (when we substitute x = 4, even though it's not a valid solution), we can determine how much needs to be added to -12 to reach 12. This involves a simple arithmetic calculation. We need to find the difference between 12 and -12. This part of the problem shifts our focus from solving the equation to a more straightforward arithmetic question, providing a different perspective on the problem.

Step 5: Calculate the Difference

To find out how much is needed to reach 12 from -12, we calculate the difference: 12 - (-12). This is equivalent to 12 + 12, which equals 24. Therefore, 24 needs to be added to -12 to reach 12. This calculation answers the second part of the question and provides a clear numerical answer. Understanding how to calculate differences between numbers, especially when negative numbers are involved, is a fundamental skill in mathematics. This step reinforces this skill and provides a practical application of arithmetic principles.

Conclusion

In this article, we meticulously solved the equation -6√1x = 12 and determined that it has no real solution. We then addressed the question of how much needs to be added to the result of the equation (which would be -12 if x=4 was a valid solution) to reach 12, and we found that the answer is 24. This exploration highlights the importance of understanding the steps involved in solving algebraic equations, as well as the crucial step of verifying solutions. It also demonstrates how mathematical problems can have multiple layers, requiring both algebraic manipulation and arithmetic calculation. By breaking down the problem into smaller, manageable steps, we were able to provide a clear and comprehensive solution. This approach is valuable for tackling any mathematical problem, regardless of its complexity. Remember, the key to success in mathematics is a solid understanding of the fundamentals and a methodical approach to problem-solving.