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

In this comprehensive article, we will delve into the process of solving the rational equation (-3d)/(d^2-2d-8) + 3/(d-4) = -2/(d+2). Rational equations, which involve algebraic fractions, often require careful manipulation and consideration of potential extraneous solutions. Our goal is to find the value(s) of 'd' that satisfy this equation. To achieve this, we will break down the problem into manageable steps, explaining each process in detail. We will begin by identifying the restrictions on the variable 'd' to avoid division by zero, which is a critical step in solving rational equations. Subsequently, we will proceed to simplify the equation by finding a common denominator, combining the fractions, and solving the resulting algebraic equation. Throughout this process, we will emphasize the importance of checking for extraneous solutions, which are solutions obtained algebraically but do not satisfy the original equation. By the end of this article, you will gain a thorough understanding of how to solve rational equations, with a focus on this specific example. This knowledge is crucial for various mathematical applications, including algebra, calculus, and other related fields. So, let's embark on this mathematical journey and unravel the solution to this challenging equation.

Before we dive into the solution, let's first understand what rational equations are and the potential pitfalls they present. A rational equation is an equation that contains one or more fractions where the variable appears in the denominator. These equations require careful handling because certain values of the variable can make the denominator equal to zero, which is undefined in mathematics. Such values are called restrictions and must be excluded from the solution set. Ignoring these restrictions can lead to extraneous solutions, which are solutions obtained algebraically but do not satisfy the original equation. Therefore, the first crucial step in solving a rational equation is to identify these restrictions. In our case, the equation is (-3d)/(d^2-2d-8) + 3/(d-4) = -2/(d+2). We need to find the values of 'd' that make the denominators d^2-2d-8, d-4, and d+2 equal to zero. This involves factoring the quadratic expression and setting each factor equal to zero. By identifying and excluding these restricted values, we ensure that our solutions are valid and do not lead to division by zero. Understanding these fundamental concepts of rational equations is essential for successfully solving them and avoiding common errors. This foundation will enable us to approach the given equation with confidence and precision.

The first crucial step in solving any rational equation is to identify the values of the variable that would make any denominator equal to zero. These values are called restrictions, and they must be excluded from the solution set. In the given equation, (-3d)/(d^2-2d-8) + 3/(d-4) = -2/(d+2), we have three denominators: d^2-2d-8, d-4, and d+2. To find the restrictions, we need to set each denominator equal to zero and solve for 'd'. First, let's consider the quadratic denominator d^2-2d-8. We can factor this expression as (d-4)(d+2). Setting this equal to zero gives us (d-4)(d+2) = 0, which implies d = 4 or d = -2. Next, we look at the denominator d-4. Setting this equal to zero gives us d-4 = 0, which implies d = 4. Finally, we consider the denominator d+2. Setting this equal to zero gives us d+2 = 0, which implies d = -2. Therefore, the restrictions are d = 4 and d = -2. These values cannot be part of the solution because they would result in division by zero, making the equation undefined. Identifying these restrictions is a critical step because it helps us avoid extraneous solutions later in the solving process. By excluding these values, we ensure that any solutions we find are valid and satisfy the original equation. This careful approach is essential for solving rational equations accurately and effectively.

After identifying the restrictions, the next step in solving the rational equation (-3d)/(d^2-2d-8) + 3/(d-4) = -2/(d+2) is to find a common denominator. This allows us to combine the fractions and simplify the equation. Looking at the denominators d^2-2d-8, d-4, and d+2, we can see that d^2-2d-8 can be factored as (d-4)(d+2). This means that the common denominator for all three fractions is (d-4)(d+2). Now, we need to rewrite each fraction with this common denominator. The first fraction, (-3d)/(d^2-2d-8), already has the common denominator (d-4)(d+2). The second fraction, 3/(d-4), needs to be multiplied by (d+2)/(d+2) to get the common denominator. This gives us (3(d+2))/((d-4)(d+2)). The third fraction, -2/(d+2), needs to be multiplied by (d-4)/(d-4) to get the common denominator. This gives us (-2(d-4))/((d-4)(d+2)). By rewriting each fraction with the common denominator (d-4)(d+2), we can now combine the fractions and simplify the equation. This step is crucial for eliminating the fractions and transforming the rational equation into a more manageable algebraic equation. Finding the common denominator is a fundamental technique in solving rational equations, and it sets the stage for the subsequent steps in the solution process.

Now that we have a common denominator, (d-4)(d+2), for all the fractions in the rational equation (-3d)/(d^2-2d-8) + 3/(d-4) = -2/(d+2), we can combine them. We've rewritten the equation as (-3d)/((d-4)(d+2)) + (3(d+2))/((d-4)(d+2)) = (-2(d-4))/((d-4)(d+2)). To combine the fractions, we add or subtract the numerators while keeping the common denominator. This gives us (-3d + 3(d+2))/((d-4)(d+2)) = (-2(d-4))/((d-4)(d+2)). Next, we simplify the numerators. On the left side, we distribute the 3 in 3(d+2) to get 3d + 6. So the numerator becomes -3d + 3d + 6, which simplifies to 6. On the right side, we distribute the -2 in -2(d-4) to get -2d + 8. Thus, the equation becomes 6/((d-4)(d+2)) = (-2d + 8)/((d-4)(d+2)). Now that we have a single fraction on each side of the equation with the same denominator, we can proceed to solve for 'd' by equating the numerators. This step is a significant simplification because it eliminates the fractions, transforming the equation into a linear equation. Combining fractions is a crucial technique in solving rational equations, allowing us to manipulate the equation into a form that is easier to solve. This step sets the foundation for the final steps in finding the solution.

With the fractions combined and the equation simplified to 6/((d-4)(d+2)) = (-2d + 8)/((d-4)(d+2)), we can now focus on solving for 'd'. Since the denominators are the same, we can equate the numerators: 6 = -2d + 8. This gives us a linear equation that is much easier to solve. To isolate 'd', we first subtract 8 from both sides of the equation: 6 - 8 = -2d + 8 - 8, which simplifies to -2 = -2d. Next, we divide both sides by -2 to solve for 'd': -2 / -2 = -2d / -2, which gives us d = 1. However, it is crucial to remember the restrictions we identified earlier: d cannot be 4 or -2. Since our solution, d = 1, is not among these restricted values, it is a potential solution. To ensure that d = 1 is indeed a valid solution, we must substitute it back into the original equation and verify that it satisfies the equation. This step is essential to rule out any extraneous solutions that may have arisen during the algebraic manipulation. Solving the equation by equating the numerators is a key step in finding the value of the variable, but it is equally important to consider the restrictions and check the solution to ensure its validity.

After finding a potential solution, it is crucial to check for extraneous solutions. These are solutions that are obtained algebraically but do not satisfy the original equation. This often happens in rational equations due to the restrictions on the variable. In our case, we found d = 1 as a potential solution for the equation (-3d)/(d^2-2d-8) + 3/(d-4) = -2/(d+2). To check if d = 1 is a valid solution, we substitute it back into the original equation. Substituting d = 1, we get: (-3(1))/((1)^2-2(1)-8) + 3/(1-4) = -2/(1+2). This simplifies to: -3/(1-2-8) + 3/(-3) = -2/3. Further simplification gives us: -3/(-9) - 1 = -2/3. This reduces to: 1/3 - 1 = -2/3. Combining the terms on the left side, we get: (1 - 3)/3 = -2/3, which simplifies to: -2/3 = -2/3. Since the equation holds true, d = 1 is indeed a valid solution and not an extraneous one. This step of checking for extraneous solutions is paramount in solving rational equations. It ensures that the solution we have found is correct and satisfies the original equation without leading to any undefined operations, such as division by zero. By verifying the solution, we can confidently conclude that d = 1 is the solution to the given rational equation.

In conclusion, the solution to the rational equation (-3d)/(d^2-2d-8) + 3/(d-4) = -2/(d+2) is d = 1. We arrived at this solution by following a systematic approach that included identifying restrictions, finding a common denominator, combining fractions, solving the resulting linear equation, and crucially, checking for extraneous solutions. The process began with recognizing the restrictions on 'd' to avoid division by zero, which led us to exclude d = 4 and d = -2. We then found the common denominator (d-4)(d+2), rewrote each fraction with this denominator, and combined the fractions. This simplification allowed us to equate the numerators and solve the linear equation, yielding a potential solution of d = 1. Finally, we rigorously checked this solution by substituting it back into the original equation, confirming that it is a valid solution and not an extraneous one. This comprehensive approach highlights the importance of each step in solving rational equations, from identifying restrictions to verifying solutions. By understanding and applying these techniques, you can confidently tackle similar problems and ensure accurate results. Rational equations are a fundamental concept in algebra, and mastering their solution is essential for further studies in mathematics and related fields. Therefore, the careful and methodical approach demonstrated in this article serves as a valuable guide for solving these types of equations.

Final Answer: The final answer is (C) d=1