Solving 7A² = 481 Find The Value Of A

In this article, we will delve into the process of solving the equation 7A² = 481 for the variable A. This is a fundamental problem in algebra that involves isolating the variable and finding its value(s). We will break down the steps involved in solving this equation, providing a clear and concise explanation for each step. Additionally, we will discuss the underlying mathematical principles and concepts that are essential for understanding the solution. This guide is designed to be comprehensive, ensuring that readers of all levels can grasp the methodology and apply it to similar problems. Whether you are a student learning algebra or someone looking to refresh your math skills, this article will provide you with the necessary tools to solve such equations effectively.

Understanding the Equation: 7A² = 481

Before we jump into the solution, let's first understand the equation we are dealing with. The equation 7A² = 481 is a quadratic equation in the form of ax² = c, where a is 7, x is A, and c is 481. In this type of equation, the variable A is squared, which means there can be two possible solutions: a positive value and a negative value. The goal is to isolate A on one side of the equation to find these values. This involves performing algebraic operations on both sides of the equation to maintain balance and ultimately determine the value(s) of A that satisfy the equation. It is crucial to understand the structure of the equation to apply the correct steps and arrive at the accurate solutions. The process of isolating the variable involves reversing the operations performed on it, ensuring that each step maintains the equation's equality. Understanding this fundamental concept is key to solving not just this equation, but a wide range of algebraic problems.

Step 1: Isolating A²

Our first step in solving the equation 7A² = 481 is to isolate the term A². To do this, we need to get rid of the coefficient 7 that is multiplying A². The key principle here is to perform the inverse operation. Since 7 is multiplying A², we need to divide both sides of the equation by 7. This maintains the balance of the equation, which is a fundamental rule in algebra. When we divide both sides by 7, the equation transforms from 7A² = 481 to A² = 481/7. This step is crucial because it simplifies the equation and brings us closer to isolating A. By isolating A², we set the stage for the next step, which involves taking the square root to solve for A. This initial step is a clear example of how algebraic manipulations are used to simplify equations and make them solvable. It's important to remember that whatever operation is performed on one side of the equation must also be performed on the other side to maintain equality.

Step 2: Calculate 481/7

After dividing both sides of the equation by 7, we have A² = 481/7. The next step is to calculate the value of 481/7. Performing this division gives us approximately 68.71. So, our equation now looks like this: A² ≈ 68.71. This step simplifies the equation further by providing a numerical value for the right side, making it easier to proceed with isolating A. This calculation is a straightforward arithmetic operation, but it is essential for moving forward in the solution process. By converting the fraction 481/7 into its decimal equivalent, we make the equation more manageable for the next step, which involves taking the square root. This simplification highlights the importance of basic arithmetic skills in solving algebraic equations. The approximation to two decimal places, as requested in the instructions, ensures that the final answer is accurate to the specified level of precision.

Step 3: Taking the Square Root

Now that we have A² ≈ 68.71, we need to find the value of A. Since A is squared, the inverse operation to undo the square is taking the square root. It's crucial to remember that when we take the square root of a number, we get two possible solutions: a positive root and a negative root. This is because both the positive and negative values, when squared, will result in the same positive number. Therefore, we need to consider both possibilities. Applying the square root to both sides of the equation, we get A = ±√68.71. This step is a critical step in solving quadratic equations because it reveals the two potential solutions for the variable. The ± symbol indicates that we have both a positive and a negative root, reflecting the symmetrical nature of squaring a number. This understanding of the properties of square roots is fundamental in algebra and is essential for solving equations involving squared variables.

Step 4: Calculating the Square Root of 68.71

We have now reached the stage where A = ±√68.71. The next step is to calculate the square root of 68.71. Using a calculator, we find that the square root of 68.71 is approximately 8.29. Therefore, we have two possible values for A: A ≈ +8.29 and A ≈ -8.29. This step provides the numerical values for A, which are the solutions to the original equation. Calculating the square root is a straightforward process using a calculator, but it is a crucial step in determining the final answers. The approximation to two decimal places, as specified in the instructions, ensures that the answers are accurate to the required level of precision. This calculation demonstrates the practical application of mathematical tools to find numerical solutions to algebraic problems.

Step 5: Final Solutions and Verification

In the previous steps, we found two possible solutions for A: A ≈ 8.29 and A ≈ -8.29. These are the values that satisfy the original equation, 7A² = 481. It is always a good practice to verify these solutions by plugging them back into the original equation to ensure they hold true. Let's first verify A ≈ 8.29: 7 * (8.29)² ≈ 7 * 68.7241 ≈ 481.0687, which is very close to 481. Now, let's verify A ≈ -8.29: 7 * (-8.29)² ≈ 7 * 68.7241 ≈ 481.0687, which is also very close to 481. The slight difference from 481 is due to rounding to two decimal places during the calculation. Since both values, approximately 8.29 and -8.29, satisfy the equation, they are the solutions. The process of verification is an essential step in problem-solving, as it confirms the accuracy of the solutions and helps to catch any errors that may have occurred during the process. This step reinforces the understanding of the equation and the solution process.

Conclusion: The Values of A

In conclusion, we have successfully solved the equation 7A² = 481 for the variable A. By following a step-by-step approach, we first isolated A² by dividing both sides of the equation by 7. We then calculated the value of 481/7, which is approximately 68.71. Next, we took the square root of both sides to find the values of A, remembering to consider both the positive and negative roots. This gave us A = ±√68.71. Finally, we calculated the square root of 68.71, which is approximately 8.29. Therefore, the two solutions for A are approximately 8.29 and -8.29. We also verified these solutions by plugging them back into the original equation, confirming their accuracy. This process demonstrates the fundamental principles of solving quadratic equations and highlights the importance of algebraic manipulation, arithmetic calculations, and attention to detail. By understanding these principles and practicing these steps, you can confidently solve similar equations and expand your mathematical skills. The solutions A ≈ 8.29 and A ≈ -8.29 are the values that satisfy the given equation, providing a complete and accurate answer to the problem.

This comprehensive guide has walked you through each step of solving the equation 7A² = 481. By understanding the underlying mathematical principles and practicing these steps, you can confidently tackle similar problems in algebra and beyond.