Solving for x2 in Advanced Algebraic Equations: A Guide

Algebraic equations can sometimes be quite challenging, but understanding the underlying principles can greatly simplify the problem-solving process. In this article, we will explore a specific problem involving the expression x2 when given the equation 3x - 1 36. The goal is to break down the solution into clear, manageable steps and provide insights that will help you tackle similar problems with confidence.

Understanding the Given Equation

The equation we are working with is 3x - 1 36. Our objective is to find the value of x and subsequently x2. Let's begin by isolating x on one side of the equation.

Step 1: Isolate x

First, add 1 to both sides of the equation to eliminate the constant:

3x - 1 1 36 1

3x 37

Step 2: Solve for x

Next, divide both sides of the equation by 3 to isolate x:

3x / 3 37 / 3

x (frac{37}{3})

This gives us the value of x. Now, we need to find x2.

Step 3: Calculate x2

To find x2, square the value of x. Let's do the calculation step-by-step:

(left(frac{37}{3}right)2 frac{372}{32} frac{1369}{9})

The value of x2 is therefore (frac{1369}{9}).

Alternative Interpretations

The solution provided above is based on a straightforward approach to solving the equation. However, there are alternative interpretations and alternative methods to solve the same problem, which may lead to different answers. Here are a couple of such approaches:

Method 1: Inverted x Term

In some contexts, the term may be interpreted as (frac{3}{x}) rather than 3x. Given the equation 3x^{-1} 36, we proceed as follows:

Step 1: Isolate x^{-1}

First, multiply both sides by x to isolate x^{-1}:

3x^{-1} cdot x 36 cdot x

3 cdot 1 36x

3 36x

Step 2: Solve for x

Next, divide both sides by 36 to isolate x:

3 / 36 x

x (frac{1}{12})

Step 3: Calculate x2

Square the value of x to find x2:

(left(frac{1}{12}right)2 frac{1}{144})

The value of x2 is therefore (frac{1}{144}).

Method 2: Contextual Interpretation

Another possible interpretation involves context-specific scenarios. For example, if the problem is part of a word problem, the interpretation might affect the result. If the original problem was intended to involve fractions, the correct interpretation would be (frac{3}{x}), leading to the answer as calculated in the second method.

Conclusion

The solution to the problem of finding x2 in the equation 3x - 1 36 depends on the interpretation of the terms involved. The straightforward method gives us (frac{1369}{9}), while the alternative method leads to (frac{1}{144}). Both approaches are valid and should be considered depending on the context and the specific interpretation of the problem.

Related Keywords

Algebraic equations Solving x2 Advanced algebra

Further Reading

For more detailed information on solving algebraic equations and understanding the nuances of algebra, consider exploring the following resources:

MathIsFun: Algebra Khan Academy: Algebra StudyPug: Algebra

By continuing to practice and explore these topics, you can build a strong foundation in algebra and advanced algebraic concepts.