Solving (2/(x-4)) + ((x+3)/x) = 8/(x^2-4x) A Step-by-Step Guide

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Introduction

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the step-by-step process of solving the given equation: {\frac{2}{x-4} + \frac{x+3}{x} = rac{8}{x^2 - 4x}}. This equation involves fractions with algebraic expressions in the denominators, making it a classic example of a rational equation. Mastering the techniques to solve such equations is crucial for various applications in algebra, calculus, and beyond. This comprehensive guide will not only provide the solution(s) but also offer a detailed explanation of each step involved, ensuring a thorough understanding of the underlying concepts. Understanding the solution to equations like {\frac{2}{x-4} + \frac{x+3}{x} = rac{8}{x^2 - 4x}} is a cornerstone of algebraic proficiency, and this article aims to empower you with the knowledge and skills to tackle similar problems confidently. By the end of this discussion, you will be able to approach rational equations with a clear strategy and execute the necessary algebraic manipulations with precision.

1. Identifying the Domain and Restrictions

Before diving into the algebraic manipulations, it's imperative to identify the domain of the equation and any potential restrictions on the variable x. The domain represents the set of all possible values of x for which the equation is defined. In this case, we have fractions, and the denominators cannot be equal to zero. Thus, we need to find the values of x that would make any of the denominators zero and exclude them from the domain. The denominators in our equation are x - 4, x, and x² - 4x. Setting each of these equal to zero, we get:

  • x - 4 = 0 => x = 4
  • x = 0
  • x² - 4x = 0 => x(x - 4) = 0 => x = 0 or x = 4

Therefore, the values x = 0 and x = 4 make the denominators zero, and they must be excluded from the domain. This is a crucial first step, as any solution we obtain later must be checked against these restrictions. Identifying the domain at the outset prevents us from accepting extraneous solutions that might arise during the algebraic manipulations. Understanding the restrictions on the variable x is fundamental to the integrity of the solution process. Failing to acknowledge these limitations can lead to incorrect conclusions and a misunderstanding of the equation's behavior. In summary, the domain of our equation is all real numbers except 0 and 4. We can express this mathematically as: x ≠ 0 and x ≠ 4. These restrictions will be critical when we evaluate the potential solutions later in the process.

2. Finding the Least Common Denominator (LCD)

To effectively combine the fractions in the equation, we need to find the least common denominator (LCD). The LCD is the smallest expression that is divisible by each of the denominators in the equation. In our case, the denominators are x - 4, x, and x² - 4x. Notice that x² - 4x can be factored as x(x - 4). This factorization reveals that the LCD must include both x and x - 4 as factors. Therefore, the LCD is x(x - 4). Finding the Least Common Denominator (LCD) is a pivotal step in solving rational equations, as it allows us to eliminate the fractions and work with a simpler algebraic expression. The LCD acts as a common ground, enabling us to combine terms and simplify the equation. To determine the LCD, we carefully examine each denominator and identify the factors involved. In this specific equation, recognizing that x² - 4x can be factored into x(x - 4) is key to correctly identifying the LCD. The LCD, x(x - 4), encompasses all the unique factors present in the denominators, ensuring that each fraction can be manipulated to have this common denominator. This process not only simplifies the equation but also provides a systematic approach to solving similar problems. Once we have the LCD, we can proceed to multiply both sides of the equation by it, effectively clearing the fractions and paving the way for further algebraic simplification. This step is essential for transforming the equation into a more manageable form that can be readily solved using standard algebraic techniques.

3. Multiplying Both Sides by the LCD

Now that we have identified the LCD as x(x - 4), the next step is to multiply both sides of the equation by this expression. This crucial step eliminates the fractions, transforming the equation into a more manageable form. Multiplying both sides of the equation {\frac{2}{x-4} + \frac{x+3}{x} = rac{8}{x^2 - 4x}} by x(x - 4), we get:

  • x(x - 4) 2x−4{\frac{2}{x-4}} + x(x - 4) x+3x{\frac{x+3}{x}} = x(x - 4) 8x2−4x{\frac{8}{x^2 - 4x}}

This multiplication distributes the LCD to each term in the equation. Now, we can simplify each term by canceling out the common factors. In the first term, (x - 4) cancels out, leaving 2x. In the second term, x cancels out, leaving (x + 3)(x - 4). On the right-hand side, x(x - 4) cancels out completely, leaving just 8. The equation now becomes:

  • 2x + (x + 3)(x - 4) = 8

This resulting equation is a quadratic equation, which we can solve using standard algebraic techniques. Multiplying both sides by the LCD is a critical step in the process of solving rational equations. It effectively clears the fractions, transforming the equation into a more familiar algebraic form. This transformation simplifies the subsequent steps, making it easier to solve for the variable. The careful distribution and cancellation of factors are essential for accurate simplification. This process highlights the importance of understanding algebraic manipulation and the properties of equations. By eliminating the fractions, we can apply standard techniques for solving polynomial equations, such as factoring, using the quadratic formula, or completing the square. The resulting equation, in this case, a quadratic equation, is significantly easier to handle, allowing us to proceed towards finding the solution(s) for x. This step underscores the power of algebraic manipulation in transforming complex equations into simpler, solvable forms.

4. Simplifying and Rearranging the Equation

With the fractions eliminated, we now have the equation: 2x + (x + 3)(x - 4) = 8. The next step is to simplify and rearrange this equation into a standard form, typically a quadratic equation in the form ax² + bx + c = 0. First, we expand the product (x + 3)(x - 4):

  • (x + 3)(x - 4) = x² - 4x + 3x - 12 = x² - x - 12

Now, substitute this back into the equation:

  • 2x + x² - x - 12 = 8

Combine like terms:

  • x² + x - 12 = 8

To get the standard quadratic form, subtract 8 from both sides:

  • x² + x - 20 = 0

This is now a quadratic equation in the standard form ax² + bx + c = 0, where a = 1, b = 1, and c = -20. Simplifying and Rearranging the Equation is a crucial step in solving for x. This process involves expanding products, combining like terms, and rearranging the equation into a standard form. For rational equations that lead to quadratic equations, this typically means transforming the equation into the form ax² + bx + c = 0. This standard form is essential because it allows us to apply well-established methods for solving quadratic equations, such as factoring, using the quadratic formula, or completing the square. The accuracy of this step is paramount, as any errors in expansion, combination, or rearrangement will propagate through the rest of the solution process. By carefully simplifying and rearranging the equation, we set the stage for efficiently and accurately finding the solution(s) for x. This step exemplifies the importance of meticulous algebraic manipulation in achieving a solvable equation. The quadratic equation x² + x - 20 = 0 is now in a format that is readily amenable to standard solution techniques.

5. Solving the Quadratic Equation

We now have the quadratic equation x² + x - 20 = 0. To solve this equation, we can use several methods, such as factoring, the quadratic formula, or completing the square. In this case, factoring is the most straightforward approach. We are looking for two numbers that multiply to -20 and add to 1. These numbers are 5 and -4. Therefore, we can factor the quadratic equation as:

  • (x + 5)(x - 4) = 0

Setting each factor equal to zero, we get:

  • x + 5 = 0 => x = -5
  • x - 4 = 0 => x = 4

Thus, we have two potential solutions: x = -5 and x = 4. Solving the Quadratic Equation is a pivotal step in determining the possible values of x that satisfy the original equation. There are various methods for solving quadratic equations, each with its own advantages and applicability depending on the specific equation. Factoring, as employed in this case, is often the quickest method when the quadratic expression can be readily factored. The quadratic formula is a universal method that can be applied to any quadratic equation, while completing the square is useful for understanding the structure of the quadratic and for deriving the quadratic formula itself. The choice of method depends on the equation's characteristics and the solver's preference. In the given equation, x² + x - 20 = 0, factoring provides a direct and efficient path to the solutions. The identification of the factors (x + 5) and (x - 4) leads to the potential solutions x = -5 and x = 4. However, it is crucial to remember that these are only potential solutions until they are checked against the original equation's domain and any restrictions.

6. Checking for Extraneous Solutions

Recall that we identified restrictions on the domain of the original equation. We found that x cannot be 0 or 4 because these values would make the denominators zero. We obtained two potential solutions: x = -5 and x = 4. However, x = 4 violates the restriction we identified earlier. Therefore, x = 4 is an extraneous solution and must be discarded. The only remaining solution is x = -5. To verify this solution, we substitute x = -5 back into the original equation:

  • 2−5−4+−5+3−5=8(−5)2−4(−5){\frac{2}{-5-4} + \frac{-5+3}{-5} = \frac{8}{(-5)^2 - 4(-5)}}*
  • 2−9+−2−5=825+20{\frac{2}{-9} + \frac{-2}{-5} = \frac{8}{25 + 20}}*
  • −29+25=845{-\frac{2}{9} + \frac{2}{5} = \frac{8}{45}}*

To check if this is true, find a common denominator, which is 45:

  • −1045+1845=845{-\frac{10}{45} + \frac{18}{45} = \frac{8}{45}}*
  • 845=845{\frac{8}{45} = \frac{8}{45}}*

The equation holds true, so x = -5 is a valid solution. Checking for Extraneous Solutions is an indispensable step in solving rational equations. Extraneous solutions are potential solutions that arise during the algebraic manipulation process but do not satisfy the original equation. These solutions typically occur because operations like squaring or multiplying by an expression containing the variable can introduce solutions that were not present in the original equation. The domain restrictions, identified at the outset, play a crucial role in this step. Any potential solution that violates these restrictions must be discarded. In this case, we identified x = 4 as a potential solution, but it was excluded because it would make the denominators in the original equation equal to zero. The remaining solution, x = -5, was then verified by substituting it back into the original equation. This verification step ensures that the solution is valid and that no algebraic errors were made during the solution process. The successful verification of x = -5 confirms that it is the only valid solution to the equation. This rigorous checking process underscores the importance of a comprehensive approach to solving rational equations, where attention to detail and adherence to established mathematical principles are paramount.

7. Conclusion

In conclusion, the solution to the equation {\frac{2}{x-4} + \frac{x+3}{x} = rac{8}{x^2 - 4x}} is x = -5. We arrived at this solution by systematically following a series of steps: identifying the domain restrictions, finding the least common denominator, multiplying both sides by the LCD, simplifying and rearranging the equation, solving the resulting quadratic equation, and, crucially, checking for extraneous solutions. This comprehensive approach ensures that we obtain the correct solution and avoid accepting any extraneous solutions that may arise during the algebraic manipulation process. The solution to the equation, x = -5, represents the value of x that satisfies the original rational equation while adhering to its domain restrictions. The step-by-step process outlined in this article provides a clear and methodical approach to solving rational equations, emphasizing the importance of each step in ensuring an accurate solution. The initial identification of domain restrictions, such as x ≠ 0 and x ≠ 4, set the stage for a rigorous solution process. Finding the LCD, x(x - 4), allowed us to eliminate the fractions and transform the equation into a more manageable form. The subsequent simplification and rearrangement led to the quadratic equation x² + x - 20 = 0, which was solved by factoring. The potential solutions, x = -5 and x = 4, were then carefully checked for extraneous solutions. The exclusion of x = 4 due to domain restrictions highlighted the necessity of this step. Finally, the verification of x = -5 by substitution into the original equation confirmed its validity. This methodical approach underscores the significance of a systematic and rigorous problem-solving strategy in mathematics.