Solving 7 - ³√(2 - X) = 12 A Step-by-Step Guide

This article delves into the step-by-step process of finding the solution to the equation 7 - ³√(2 - x) = 12. We'll break down the equation, isolate the variable, and explore the underlying mathematical principles involved. Whether you're a student grappling with algebra or simply someone who enjoys a good mathematical puzzle, this guide will provide a clear and concise path to the solution. Let's embark on this mathematical journey together and uncover the value of 'x' that satisfies this intriguing equation.

H2: Understanding the Equation and the Goal

Before we jump into the solution, let's take a moment to understand the equation itself. The equation 7 - ³√(2 - x) = 12 involves a cube root, a subtraction, and a constant. Our goal is to isolate the variable 'x' on one side of the equation to determine its value. To achieve this, we'll employ a series of algebraic manipulations, carefully undoing the operations that are applied to 'x'.

First, it's crucial to recognize the order of operations. The cube root operation applies to the expression (2 - x). The result of this cube root is then subtracted from 7. Finally, the entire expression is set equal to 12. To solve for 'x', we essentially need to reverse these operations in the opposite order.

The strategy involves isolating the cube root term first. This can be achieved by subtracting 7 from both sides of the equation. This step maintains the equality while bringing us closer to isolating the cube root. Then, we'll need to deal with the negative sign in front of the cube root. Multiplying both sides by -1 will accomplish this. Once the cube root term is isolated, we can eliminate the cube root by cubing both sides of the equation. This is the inverse operation of taking the cube root and will reveal the expression inside the root. Finally, we'll use basic algebraic manipulation to isolate 'x' and arrive at the solution.

Throughout this process, it's essential to remember the fundamental principles of algebra: whatever operation we perform on one side of the equation, we must perform on the other side to maintain the balance. This ensures that the equation remains valid and that the solution we obtain is accurate. Let's now move on to the step-by-step solution process, where we'll see these principles in action.

H2: Step-by-Step Solution

Let's dive into the step-by-step solution of the equation 7 - ³√(2 - x) = 12. Each step is carefully explained to ensure clarity and understanding.

H3: Step 1: Isolate the Cube Root Term

Our initial goal is to isolate the cube root term, ³√(2 - x). To do this, we need to get rid of the 7 on the left side of the equation. We achieve this by subtracting 7 from both sides of the equation:

7 - ³√(2 - x) - 7 = 12 - 7

This simplifies to:

-³√(2 - x) = 5

Now, the cube root term is almost isolated, but it has a negative sign in front of it. We need to address this in the next step.

H3: Step 2: Eliminate the Negative Sign

To eliminate the negative sign in front of the cube root, we multiply both sides of the equation by -1:

-1 * (-³√(2 - x)) = -1 * 5

This gives us:

³√(2 - x) = -5

Now, the cube root term is completely isolated on the left side of the equation. We are ready to move on to the next step, which involves eliminating the cube root itself.

H3: Step 3: Eliminate the Cube Root

The cube root can be eliminated by performing the inverse operation, which is cubing. We cube both sides of the equation:

(³√(2 - x))³ = (-5)³

This simplifies to:

2 - x = -125

We have now successfully removed the cube root, and we are left with a simple linear equation. The next step is to isolate 'x'.

H3: Step 4: Isolate 'x'

To isolate 'x', we first subtract 2 from both sides of the equation:

2 - x - 2 = -125 - 2

This simplifies to:

-x = -127

Finally, to get 'x' by itself, we multiply both sides of the equation by -1:

-1 * (-x) = -1 * (-127)

This gives us the solution:

x = 127

Therefore, the solution to the equation 7 - ³√(2 - x) = 12 is x = 127. We have successfully navigated through the algebraic manipulations to arrive at the final answer.

H2: Verification of the Solution

It's always a good practice to verify the solution we've obtained to ensure its accuracy. To do this, we substitute the value of x back into the original equation and check if it holds true.

Substituting x = 127 into the original equation 7 - ³√(2 - x) = 12, we get:

7 - ³√(2 - 127) = 12

Now, let's simplify the expression step-by-step:

7 - ³√(-125) = 12

The cube root of -125 is -5, so we have:

7 - (-5) = 12

7 + 5 = 12

12 = 12

Since the equation holds true, our solution x = 127 is verified. This confirms that we have correctly solved the equation and found the accurate value of 'x'. Verification is a crucial step in problem-solving, as it helps to catch any potential errors and ensures the reliability of the solution.

H2: Alternative Methods and Approaches

While we've presented a clear step-by-step solution, it's worth noting that there might be alternative approaches to solving the equation 7 - ³√(2 - x) = 12. Different individuals may prefer different methods based on their understanding and comfort level with algebraic manipulations. Let's briefly explore a possible alternative approach.

One alternative could involve rearranging the equation slightly before isolating the cube root. Instead of subtracting 7 from both sides in the first step, one could add ³√(2 - x) to both sides and subtract 12 from both sides. This would lead to the equation:

7 - 12 = ³√(2 - x)

-5 = ³√(2 - x)

This alternative first isolates the cube root on the right-hand side, which some individuals might find more intuitive. From this point, the remaining steps would be similar to the original solution: cubing both sides, isolating 'x', and arriving at the same solution, x = 127.

The key takeaway here is that mathematical problems often have multiple solution paths. While the fundamental principles of algebra remain the same, the order in which we apply these principles can vary. Exploring different approaches not only enhances our problem-solving skills but also deepens our understanding of the underlying mathematical concepts.

H2: Common Mistakes to Avoid

When solving equations involving radicals, such as the equation 7 - ³√(2 - x) = 12, it's crucial to be aware of common mistakes that can lead to incorrect solutions. Identifying and avoiding these pitfalls can significantly improve accuracy and confidence in problem-solving. Let's discuss some of these common mistakes:

• Incorrectly applying the order of operations: As we emphasized earlier, the order of operations is paramount. A frequent error is to incorrectly distribute the negative sign or to attempt to eliminate the cube root before isolating it. Always remember to isolate the radical term before raising both sides to a power.

• Forgetting to cube the negative sign: When cubing a negative number, the result is also negative. For instance, (-5)³ = -125. Some individuals may mistakenly calculate this as 125. Paying close attention to the signs is essential.

• Making errors in algebraic manipulation: Simple arithmetic errors, such as adding or subtracting incorrectly, can derail the entire solution process. It's always wise to double-check each step and ensure accuracy in the algebraic manipulations.

• Not verifying the solution: As we demonstrated, verifying the solution is a crucial step. It helps to identify any potential errors made during the solving process. Skipping this step can lead to accepting an incorrect solution.

By being mindful of these common mistakes and taking the necessary precautions, you can minimize the chances of error and confidently solve equations involving radicals.

H2: Conclusion

In conclusion, we have successfully solved the equation 7 - ³√(2 - x) = 12 by employing a step-by-step approach. We began by isolating the cube root term, then eliminated the negative sign, cubed both sides of the equation, and finally isolated 'x' to arrive at the solution x = 127. We also verified the solution to ensure its accuracy. Furthermore, we explored an alternative solution method and discussed common mistakes to avoid when solving equations involving radicals.

Understanding the underlying principles of algebra, such as the order of operations and the importance of maintaining equality, is crucial for solving mathematical problems effectively. Practice and attention to detail are key to mastering these skills and confidently tackling more complex equations. We hope this comprehensive guide has provided you with a clear understanding of the solution process and enhanced your problem-solving abilities. Remember, mathematics is a journey of exploration and discovery, and each solved equation brings us one step closer to a deeper understanding of the world around us.