Simplify And Solve The Equation E(x) = (1/(x^2-4x) + 2/(16-x^2) + 1/(16+4x)) / ((x+4)/(x-4))^-1.

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Introduction

In this comprehensive guide, we will delve into the intricate process of simplifying and solving a complex algebraic expression. Our focus is on the expression E(x) = (1/(x^2-4x) + 2/(16-x^2) + 1/(16+4x)) / ((x+4)/(x-4))^-1, which presents a fascinating challenge in the realm of mathematics. This detailed exploration is designed to provide a step-by-step understanding, making it accessible for students and enthusiasts alike. We'll cover essential algebraic techniques, including factorization, finding common denominators, simplifying rational expressions, and handling inverse operations. This journey will not only enhance your algebraic skills but also deepen your appreciation for the elegance and precision of mathematical manipulations. Whether you're a student grappling with algebraic concepts or simply a math aficionado seeking to refine your skills, this guide promises to be an invaluable resource. Let's embark on this mathematical adventure together, unraveling the complexities of this expression and emerging with a clearer understanding of the underlying principles at play.

Breaking Down the Expression: A Step-by-Step Approach

The given expression is E(x) = (1/(x^2-4x) + 2/(16-x^2) + 1/(16+4x)) / ((x+4)/(x-4))^-1. To effectively simplify this, we'll dissect it into manageable parts, addressing each component systematically. The initial step involves simplifying the terms within the parentheses in the numerator. This requires a keen understanding of factorization and the ability to identify common denominators. We will then tackle the division by a rational expression, which is essentially multiplication by its reciprocal. Throughout this process, we will emphasize the importance of meticulousness and attention to detail, as even a minor error can significantly alter the outcome. Our approach is not just about arriving at the correct answer but also about fostering a deep understanding of the underlying mathematical principles. By breaking down the problem into smaller, more digestible steps, we aim to make the process less daunting and more accessible. This methodical approach will not only aid in solving this particular expression but also equip you with the skills necessary to tackle similar algebraic challenges in the future.

Step 1: Factor the Denominators

The first key step in simplifying our expression is to factor the denominators of each fraction. This is crucial for identifying common denominators and combining the fractions effectively. Let's examine each denominator individually:

  1. x^2 - 4x: This can be factored by taking out the common factor, which is x. So, x^2 - 4x = x(x - 4).
  2. 16 - x^2: This is a difference of squares, which factors into (4 - x)(4 + x). Note that this can also be written as -(x - 4)(x + 4).
  3. 16 + 4x: Here, the common factor is 4. Factoring it out gives us 4(4 + x).

By factoring these denominators, we've laid the groundwork for finding a common denominator, a crucial step in simplifying the expression. This meticulous approach to factorization is a cornerstone of algebraic manipulation, allowing us to transform complex expressions into more manageable forms. The ability to quickly and accurately factor expressions is a valuable skill in mathematics, and this step serves as a practical demonstration of its importance.

Step 2: Find the Least Common Denominator (LCD)

After factoring the denominators, the next crucial step is to determine the Least Common Denominator (LCD). The LCD is essential for combining fractions, as it allows us to express each fraction with a common base, making addition and subtraction straightforward. In our case, the factored denominators are x(x - 4), -(x - 4)(x + 4), and 4(4 + x). To find the LCD, we need to identify all unique factors and their highest powers present in the denominators. Looking at our factored forms, we can identify the following unique factors: x, (x - 4), and (x + 4). The highest power of each factor is 1, as they each appear only once in any single denominator. Therefore, the LCD is the product of these unique factors, which is 4x(x - 4)(x + 4). It's worth noting that the factor of 4 from the last denominator needs to be included in the LCD. Understanding how to find the LCD is a fundamental skill in algebra, and it's a key step in simplifying complex expressions like the one we're working with.

Step 3: Rewrite Fractions with the LCD

Now that we've determined the LCD as 4x(x - 4)(x + 4), our next task is to rewrite each fraction in the original expression using this common denominator. This involves multiplying both the numerator and the denominator of each fraction by the necessary factors to achieve the LCD. Let's take each fraction one by one:

  1. For the fraction 1/(x(x - 4)), we need to multiply both the numerator and the denominator by 4(x + 4) to get the LCD. This gives us (4(x + 4)) / (4x(x - 4)(x + 4)).
  2. For the fraction 2/(-(x - 4)(x + 4)), we need to multiply both the numerator and the denominator by 4x to get the LCD. This results in (2 * 4x) / (-4x(x - 4)(x + 4)), which simplifies to 8x / (-4x(x - 4)(x + 4)). To make the denominator positive and consistent with our LCD, we can rewrite this as (-8x) / (4x(x - 4)(x + 4)).
  3. For the fraction 1/(4(4 + x)), we need to multiply both the numerator and the denominator by x(x - 4) to achieve the LCD. This yields (x(x - 4)) / (4x(x - 4)(x + 4)).

By rewriting each fraction with the LCD, we've created a common ground for combining these fractions, which is the next step in simplifying our expression. This process highlights the importance of understanding equivalent fractions and the role of the LCD in algebraic manipulations.

Step 4: Combine the Fractions

With all fractions now expressed using the Least Common Denominator (LCD) of 4x(x - 4)(x + 4), we can proceed to combine them. This involves adding the numerators while keeping the common denominator. Our rewritten fractions are:

  1. (4(x + 4)) / (4x(x - 4)(x + 4))
  2. (-8x) / (4x(x - 4)(x + 4))
  3. (x(x - 4)) / (4x(x - 4)(x + 4))

Adding the numerators, we get 4(x + 4) - 8x + x(x - 4). Let's simplify this expression:

  • Expand 4(x + 4) to get 4x + 16.
  • Expand x(x - 4) to get x^2 - 4x.
  • Combine the terms: 4x + 16 - 8x + x^2 - 4x.
  • Simplify to get x^2 - 8x + 16.

So, the combined fraction is (x^2 - 8x + 16) / (4x(x - 4)(x + 4)). This step demonstrates the power of using a common denominator to simplify complex expressions. By combining the fractions into a single term, we've made significant progress towards our goal of simplifying the original expression. The ability to confidently add and subtract fractions with different denominators is a fundamental skill in algebra, and this step provides a clear illustration of its application.

Step 5: Simplify the Numerator

Following the combination of fractions, our expression now contains the numerator x^2 - 8x + 16. The next crucial step is to simplify this numerator further. Upon closer inspection, we can recognize that x^2 - 8x + 16 is a perfect square trinomial. This is a significant observation because perfect square trinomials can be factored into a binomial squared, which greatly simplifies the expression. Specifically, x^2 - 8x + 16 can be factored as (x - 4)^2. This factorization is a key step in our simplification process, as it allows us to express the numerator in a more compact and manageable form. Recognizing patterns like perfect square trinomials is an invaluable skill in algebra, and it often leads to significant simplifications in complex expressions. By factoring the numerator, we're not only making the expression easier to work with but also setting the stage for potential cancellations with factors in the denominator, which will further simplify our expression.

Step 6: Invert and Multiply

Now, let's address the second part of our original expression, which involves division. We have E(x) = ((x^2 - 8x + 16) / (4x(x - 4)(x + 4))) / ((x + 4)/(x - 4))^-1. The term ((x + 4)/(x - 4))^-1 represents the inverse of the fraction (x + 4)/(x - 4). To handle division by a fraction, we multiply by its reciprocal. The reciprocal of (x + 4)/(x - 4) is (x - 4)/(x + 4). Therefore, dividing by ((x + 4)/(x - 4))^-1 is the same as multiplying by (x - 4)/(x + 4). This transformation is a fundamental concept in algebra, turning a division problem into a multiplication problem, which is often easier to handle. Our expression now becomes ((x - 4)^2) / (4x(x - 4)(x + 4)) * ((x - 4)/(x + 4)). This step is crucial because it sets up the expression for further simplification through cancellation of common factors, a technique that significantly reduces the complexity of algebraic expressions.

Step 7: Cancel Common Factors

With our expression now in the form of a multiplication of fractions, ((x - 4)^2) / (4x(x - 4)(x + 4)) * ((x - 4)/(x + 4)), we can proceed to simplify by canceling out common factors. This is a critical step in reducing the expression to its simplest form. We have the factor (x - 4) appearing multiple times in both the numerator and the denominator. Specifically, (x - 4)^2 in the numerator can be written as (x - 4)(x - 4). One of these (x - 4) terms can cancel with the (x - 4) term in the denominator of the first fraction. Additionally, we have another (x - 4) term in the numerator of the second fraction, which can cancel with another (x - 4) term in the denominator of the first fraction. We also have (x + 4) in both the numerator and the denominator, which can be canceled out. After performing these cancellations, our expression simplifies considerably. This process of canceling common factors is a cornerstone of algebraic simplification, and it's a technique that is frequently used to reduce complex expressions to their most basic form. By carefully identifying and canceling these factors, we're able to reveal the underlying structure of the expression and arrive at a more concise representation.

Step 8: Write the Simplified Expression

After meticulously canceling common factors from our expression, we've reached the final stage of simplification. Let's recap our expression after the cancellations. We started with ((x - 4)^2) / (4x(x - 4)(x + 4)) * ((x - 4)/(x + 4)). We canceled one (x - 4) term from the numerator with one from the denominator, and then we canceled the remaining (x - 4) term in the numerator of the second fraction with the (x - 4) term in the denominator of the first fraction. We also canceled the (x + 4) term. What remains is 1 / (4x). This is the simplified form of the original complex expression. The journey from the initial expression to this simplified form highlights the power of algebraic manipulation and the importance of techniques like factorization, finding common denominators, and canceling common factors. The simplified expression, 1 / (4x), is much easier to understand and work with, demonstrating the value of simplification in mathematics. This final step underscores the elegance and efficiency of mathematical processes in reducing complexity and revealing the underlying simplicity of seemingly intricate expressions.

Final Answer

In conclusion, after a detailed step-by-step simplification process, the expression E(x) = (1/(x^2-4x) + 2/(16-x^2) + 1/(16+4x)) / ((x+4)/(x-4))^-1 simplifies to 1 / (4x). This journey involved a series of algebraic techniques, including factoring denominators, finding the least common denominator, rewriting fractions, combining fractions, simplifying the numerator, inverting and multiplying, and canceling common factors. Each step was crucial in transforming the complex initial expression into its simplified form. This exercise not only demonstrates the power of algebraic manipulation but also reinforces the importance of a systematic and methodical approach to problem-solving in mathematics. The final answer, 1 / (4x), is a testament to the elegance and efficiency of mathematical simplification, showcasing how complex expressions can be reduced to their most basic form through careful and precise application of algebraic principles.