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

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In the realm of mathematics, solving equations is a fundamental skill. This article will delve into the step-by-step process of solving the equation 7 - ³√(2 - x) = 12. We will explore the underlying principles, demonstrate the algebraic manipulations involved, and provide a clear and concise explanation of the solution. This guide is designed to be accessible to learners of all levels, from students encountering this type of equation for the first time to those seeking a refresher on their algebra skills.

Understanding the Equation

The equation 7 - ³√(2 - x) = 12 involves a cube root, which adds a layer of complexity compared to simple linear equations. To effectively solve this, we need to isolate the cube root term and then eliminate it by cubing both sides of the equation. This process requires careful attention to algebraic manipulations and a solid understanding of the order of operations. Let's break down each step involved in finding the solution for x.

Step 1: Isolate the Cube Root

The first crucial step in solving this equation is to isolate the cube root term, ³√(2 - x). This means we want to get the cube root expression by itself on one side of the equation. To do this, we can subtract 7 from both sides of the equation:

7 - ³√(2 - x) - 7 = 12 - 7

This simplifies to:

-³√(2 - x) = 5

Now, to get rid of the negative sign in front of the cube root, we can multiply both sides of the equation by -1:

(-1) * -³√(2 - x) = 5 * (-1)

This gives us:

³√(2 - x) = -5

At this point, the cube root term is successfully isolated. We are now ready to proceed to the next step, which involves eliminating the cube root.

Step 2: Eliminate the Cube Root

To eliminate the cube root, we need to perform the inverse operation, which is cubing. We will cube both sides of the equation. This means raising each side of the equation to the power of 3:

(³√(2 - x))³ = (-5)³

When we cube a cube root, the root and the cube effectively cancel each other out, leaving us with the expression inside the cube root:

2 - x = (-5)³

Now we need to calculate (-5)³. This means multiplying -5 by itself three times:

(-5)³ = -5 * -5 * -5 = -125

So our equation now becomes:

2 - x = -125

We have successfully eliminated the cube root and simplified the equation to a linear form.

Step 3: Solve for x

Now that we have a linear equation, 2 - x = -125, we can solve for x. The goal is to isolate x on one side of the equation. First, we can subtract 2 from both sides:

2 - x - 2 = -125 - 2

This simplifies to:

-x = -127

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

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

This gives us:

x = 127

Therefore, the solution to the equation 7 - ³√(2 - x) = 12 is x = 127.

Verifying the Solution

It is always a good practice to verify the solution by substituting the value of x back into the original equation to ensure it holds true. Let's substitute x = 127 into the original equation:

7 - ³√(2 - 127) = 12

First, we simplify the expression inside the cube root:

2 - 127 = -125

So the equation becomes:

7 - ³√(-125) = 12

Now, we find the cube root of -125:

³√(-125) = -5

Substituting this back into the equation:

7 - (-5) = 12

Simplifying:

7 + 5 = 12

12 = 12

Since the equation holds true, our solution x = 127 is correct.

Common Mistakes to Avoid

When solving equations involving radicals, it is easy to make mistakes. Here are some common pitfalls to avoid:

  1. Incorrectly isolating the radical term: Make sure to perform the correct operations to isolate the cube root term before cubing both sides.
  2. Forgetting the negative sign: Pay close attention to negative signs, especially when cubing negative numbers. A negative number cubed remains negative.
  3. Arithmetic errors: Double-check all arithmetic operations, especially when dealing with negative numbers and exponents.
  4. Not verifying the solution: Always verify your solution by substituting it back into the original equation to ensure it is correct. This helps catch any errors made during the solving process.

By avoiding these common mistakes, you can increase your accuracy and confidence in solving equations involving radicals.

Additional Examples and Practice Problems

To further solidify your understanding, let's consider another example:

Example: Solve the equation 5 + ³√(x + 1) = 2

  1. Isolate the cube root: Subtract 5 from both sides:

    ³√(x + 1) = -3

  2. Eliminate the cube root: Cube both sides:

    (³√(x + 1))³ = (-3)³

    x + 1 = -27

  3. Solve for x: Subtract 1 from both sides:

    x = -28

Verification: Substitute x = -28 back into the original equation:

5 + ³√(-28 + 1) = 2

5 + ³√(-27) = 2

5 + (-3) = 2

2 = 2

Solution is verified.

Practice Problems:

  1. Solve: 10 - ³√(3x - 2) = 6
  2. Solve: 4 + ³√(2x + 7) = 1
  3. Solve: 8 - ³√(5 - x) = 11

Working through these practice problems will help you reinforce the concepts and techniques discussed in this article.

Conclusion

Solving equations involving cube roots requires a systematic approach and attention to detail. By following the steps outlined in this guide – isolating the cube root, eliminating the cube root by cubing, and solving the resulting linear equation – you can successfully solve these types of equations. Remember to verify your solution and avoid common mistakes to ensure accuracy. With practice and a solid understanding of algebraic principles, you can master the art of solving equations with cube roots. This skill is not only valuable in mathematics but also in various fields that require problem-solving and analytical thinking.

The equation 7 - ³√(2 - x) = 12 serves as a great example to illustrate these concepts. By understanding the step-by-step solution, you can apply these techniques to solve similar problems and enhance your mathematical proficiency. Keep practicing, and you will become more confident and skilled in solving a wide range of algebraic equations.