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

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This article provides a detailed solution to the rational equation −3dd2−2d−8+3d−4=−2d+2\frac{-3d}{d^2 - 2d - 8} + \frac{3}{d - 4} = \frac{-2}{d + 2}. Rational equations, which involve fractions with polynomials in the numerator and denominator, often appear complex, but they can be solved systematically using algebraic techniques. This guide will walk you through each step, ensuring clarity and understanding. We will cover factoring, finding the least common denominator (LCD), eliminating fractions, solving the resulting polynomial equation, and checking for extraneous solutions. Extraneous solutions are particularly important in rational equations because they can arise from multiplying both sides of the equation by an expression that equals zero.

Understanding Rational Equations

Rational equations are equations that contain one or more rational expressions. A rational expression is a fraction where the numerator and denominator are polynomials. To effectively solve rational equations, it is important to have a solid understanding of algebraic manipulations, including factoring polynomials and finding common denominators. The key strategy in solving rational equations is to eliminate the fractions by multiplying both sides of the equation by the least common denominator (LCD). This transforms the equation into a more manageable form, usually a polynomial equation, which can then be solved using standard techniques. However, this process can sometimes introduce extraneous solutions, so it is crucial to verify each solution in the original equation.

The process involves several key steps: First, factor the denominators of the rational expressions. This helps in identifying the LCD. Then, find the LCD, which is the least common multiple of all the denominators. Next, multiply both sides of the equation by the LCD to eliminate the fractions. This results in a polynomial equation. Solve the polynomial equation using factoring, the quadratic formula, or other appropriate methods. Finally, check each solution in the original rational equation to ensure it does not result in division by zero, as this would make the solution extraneous. This step is critical because multiplying by the LCD, which contains variables, can introduce values that are not valid solutions to the original equation.

Step 1: Factoring the Denominators

The first crucial step in solving the equation is to factor the denominators of the rational expressions. This will help us identify the least common denominator (LCD) and simplify the equation. Our given equation is:

−3dd2−2d−8+3d−4=−2d+2\frac{-3d}{d^2 - 2d - 8} + \frac{3}{d - 4} = \frac{-2}{d + 2}

We need to factor the quadratic denominator d2−2d−8d^2 - 2d - 8. We are looking for two numbers that multiply to -8 and add to -2. These numbers are -4 and 2. Therefore, we can factor the quadratic as follows:

d2−2d−8=(d−4)(d+2)d^2 - 2d - 8 = (d - 4)(d + 2)

Now, we rewrite the equation with the factored denominator:

−3d(d−4)(d+2)+3d−4=−2d+2\frac{-3d}{(d - 4)(d + 2)} + \frac{3}{d - 4} = \frac{-2}{d + 2}

Factoring the denominators is a critical first step because it allows us to see the structure of the equation more clearly and identify common factors. By factoring the quadratic expression, we have transformed it into a product of two linear expressions, which simplifies the process of finding the least common denominator. The ability to accurately factor polynomials is a fundamental skill in algebra and is essential for solving rational equations and other types of algebraic problems. In this case, factoring the quadratic denominator makes it evident that the LCD will involve the factors (d - 4) and (d + 2), which is crucial for the next step in solving the equation.

Step 2: Finding the Least Common Denominator (LCD)

In this section, we will identify the least common denominator (LCD) for the given rational equation. The LCD is the smallest expression that is divisible by each denominator in the equation. Once we have the LCD, we can multiply both sides of the equation by it to eliminate the fractions. Looking at our equation with the factored denominators:

−3d(d−4)(d+2)+3d−4=−2d+2\frac{-3d}{(d - 4)(d + 2)} + \frac{3}{d - 4} = \frac{-2}{d + 2}

We can see that the denominators are (d−4)(d+2)(d - 4)(d + 2), (d−4)(d - 4), and (d+2)(d + 2). To find the LCD, we need to include each factor the greatest number of times it appears in any denominator. In this case, the factors are (d−4)(d - 4) and (d+2)(d + 2). The denominator (d−4)(d+2)(d - 4)(d + 2) contains both factors once, while (d−4)(d - 4) and (d+2)(d + 2) contain each factor individually. Therefore, the LCD is the product of these factors:

LCD=(d−4)(d+2)LCD = (d - 4)(d + 2)

Identifying the LCD is a key step in solving rational equations because it allows us to clear the fractions and work with a simpler polynomial equation. The LCD is essentially the smallest expression that can be used as a common denominator for all the fractions in the equation. By multiplying each term by the LCD, we ensure that all the denominators cancel out, leaving us with an equation that is easier to solve. This process relies on the fundamental principle that multiplying both sides of an equation by the same non-zero quantity preserves the equality. In this case, the LCD is the product of the distinct linear factors present in the denominators, which makes it the most efficient common denominator to use.

Step 3: Multiplying by the LCD and Simplifying

Now that we have the LCD, which is (d−4)(d+2)(d - 4)(d + 2), we will multiply both sides of the equation by the LCD to eliminate the fractions. This step is essential for transforming the rational equation into a polynomial equation that we can solve. Starting with our equation:

−3d(d−4)(d+2)+3d−4=−2d+2\frac{-3d}{(d - 4)(d + 2)} + \frac{3}{d - 4} = \frac{-2}{d + 2}

We multiply each term by the LCD:

(d−4)(d+2)(−3d(d−4)(d+2)+3d−4)=(d−4)(d+2)(−2d+2)(d - 4)(d + 2) \left( \frac{-3d}{(d - 4)(d + 2)} + \frac{3}{d - 4} \right) = (d - 4)(d + 2) \left( \frac{-2}{d + 2} \right)

Distribute the LCD to each term:

(d−4)(d+2)⋅−3d(d−4)(d+2)+(d−4)(d+2)⋅3d−4=(d−4)(d+2)⋅−2d+2(d - 4)(d + 2) \cdot \frac{-3d}{(d - 4)(d + 2)} + (d - 4)(d + 2) \cdot \frac{3}{d - 4} = (d - 4)(d + 2) \cdot \frac{-2}{d + 2}

Now, we simplify by canceling out common factors in each term. In the first term, (d−4)(d+2)(d - 4)(d + 2) cancels out, leaving −3d-3d. In the second term, (d−4)(d - 4) cancels out, leaving 3(d+2)3(d + 2). In the third term, (d+2)(d + 2) cancels out, leaving −2(d−4)-2(d - 4). This gives us:

−3d+3(d+2)=−2(d−4)-3d + 3(d + 2) = -2(d - 4)

Next, we distribute the constants in each term:

−3d+3d+6=−2d+8-3d + 3d + 6 = -2d + 8

Multiplying both sides by the LCD is the key step in clearing the fractions from the rational equation. This transformation is based on the principle that if we perform the same operation on both sides of an equation, the equality is maintained. By multiplying each term by the LCD, we ensure that the denominators are canceled out, thus simplifying the equation. The process of simplifying after multiplying by the LCD involves distributing the LCD to each term and then canceling out common factors. This step requires careful attention to detail to ensure that all terms are correctly multiplied and simplified. The result is a polynomial equation, which is typically easier to solve than the original rational equation. This transformation allows us to apply standard algebraic techniques to find the solutions.

Step 4: Solving the Resulting Polynomial Equation

After multiplying by the LCD and simplifying, we now have a polynomial equation to solve. Our equation from the previous step is:

−3d+3d+6=−2d+8-3d + 3d + 6 = -2d + 8

First, we combine like terms on the left side of the equation:

6=−2d+86 = -2d + 8

Next, we want to isolate the term with dd. We subtract 8 from both sides of the equation:

6−8=−2d+8−86 - 8 = -2d + 8 - 8

−2=−2d-2 = -2d

Now, we divide both sides by -2 to solve for dd:

−2−2=−2d−2\frac{-2}{-2} = \frac{-2d}{-2}

1=d1 = d

So, we have found a potential solution: d=1d = 1.

Solving the resulting polynomial equation is a critical step in finding the solution to the original rational equation. Once the fractions have been cleared, we are left with a simpler equation that can be solved using standard algebraic techniques. The specific methods used to solve the polynomial equation depend on its degree and form. In this case, the equation simplifies to a linear equation, which can be solved by isolating the variable. This involves performing operations such as combining like terms, adding or subtracting constants, and multiplying or dividing both sides of the equation by a constant. The goal is to isolate the variable on one side of the equation, thus determining its value. However, it is crucial to remember that any solution obtained must be checked in the original rational equation to ensure it is not extraneous.

Step 5: Checking for Extraneous Solutions

It is crucial to check the solution we found (d=1d = 1) in the original equation to make sure it is not an extraneous solution. Extraneous solutions are solutions that satisfy the transformed equation but not the original equation. This often happens when dealing with rational equations because multiplying by the LCD can introduce values that make the denominator zero. Our original equation is:

−3dd2−2d−8+3d−4=−2d+2\frac{-3d}{d^2 - 2d - 8} + \frac{3}{d - 4} = \frac{-2}{d + 2}

Substitute d=1d = 1 into the equation:

−3(1)(1)2−2(1)−8+31−4=−21+2\frac{-3(1)}{(1)^2 - 2(1) - 8} + \frac{3}{1 - 4} = \frac{-2}{1 + 2}

−31−2−8+3−3=−23\frac{-3}{1 - 2 - 8} + \frac{3}{-3} = \frac{-2}{3}

−3−9+(−1)=−23\frac{-3}{-9} + (-1) = \frac{-2}{3}

13−1=−23\frac{1}{3} - 1 = \frac{-2}{3}

13−33=−23\frac{1}{3} - \frac{3}{3} = \frac{-2}{3}

−23=−23\frac{-2}{3} = \frac{-2}{3}

The solution d=1d = 1 checks out, meaning it is a valid solution.

Checking for extraneous solutions is a critical final step in solving rational equations. This step is necessary because multiplying both sides of the equation by the LCD can introduce solutions that are not valid for the original equation. Extraneous solutions occur when a value makes the denominator of any fraction in the original equation equal to zero, which is undefined. To check for extraneous solutions, each potential solution must be substituted back into the original equation. If the substitution results in a true statement, then the solution is valid. However, if the substitution results in a false statement or an undefined expression (such as division by zero), then the solution is extraneous and must be discarded. This step ensures that the solutions obtained are consistent with the original problem and do not violate any mathematical rules.

Conclusion

We have successfully solved the rational equation −3dd2−2d−8+3d−4=−2d+2\frac{-3d}{d^2 - 2d - 8} + \frac{3}{d - 4} = \frac{-2}{d + 2}. The steps we followed included factoring the denominators, finding the LCD, multiplying both sides by the LCD, simplifying the equation, solving the resulting polynomial equation, and checking for extraneous solutions. Our solution is d=1d = 1, which we verified by substituting it back into the original equation. Therefore, the correct answer is:

C. d = 1

In summary, solving rational equations involves a systematic approach that combines algebraic techniques with careful attention to detail. Factoring, finding the LCD, clearing fractions, solving polynomial equations, and checking for extraneous solutions are all crucial steps. By following these steps, we can effectively solve rational equations and ensure the accuracy of our solutions. Understanding and mastering these techniques is essential for success in algebra and related fields.