Solution To The Equation (-3d)/(d^2-2d-8) + 3/(d-4) = -2/(d+2)

by ADMIN 63 views

This article delves into the process of solving a rational equation, providing a comprehensive, step-by-step solution to the equation −3dd2−2d−8+3d−4=−2d+2\frac{-3 d}{d^2-2 d-8}+\frac{3}{d-4}=\frac{-2}{d+2}. Rational equations, which involve fractions with polynomials in the numerator and denominator, can seem daunting at first. However, with a systematic approach, these equations can be solved effectively. Understanding the underlying principles of algebraic manipulation, factoring, and the identification of extraneous solutions is key to mastering these problems. This guide will not only provide the solution but also explain the reasoning behind each step, enhancing your understanding of how to solve similar problems in the future. By the end of this article, you'll have a clear understanding of how to tackle rational equations and arrive at the correct solution. So, let's embark on this journey of algebraic exploration and demystify the process of solving rational equations.

Understanding Rational Equations

Before diving into the solution, it's crucial to understand what rational equations are and the fundamental principles involved in solving them. A rational equation is essentially an equation that contains one or more fractions where the numerator and denominator are polynomials. Solving these equations requires a blend of algebraic techniques, including factoring, finding common denominators, and simplifying expressions. The primary goal is to isolate the variable, but this process is often complicated by the presence of fractions. The first step in solving a rational equation often involves eliminating the fractions by multiplying both sides of the equation by the least common denominator (LCD). This step transforms the rational equation into a more manageable polynomial equation. However, it's essential to remember that multiplying by an expression containing a variable can introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, it's crucial to check each potential solution in the original equation to ensure its validity. The principles of factoring polynomials also play a significant role in solving rational equations. Factoring allows us to simplify expressions, identify common factors, and determine the LCD. Understanding different factoring techniques, such as factoring quadratic expressions and the difference of squares, is invaluable in this context. Moreover, identifying restrictions on the variable is essential. Since division by zero is undefined, we must identify values of the variable that would make any denominator in the original equation equal to zero. These values are excluded from the solution set. By grasping these fundamental concepts, you'll be well-equipped to solve a wide range of rational equations with confidence.

Step 1: Factoring the Denominator

The initial crucial step in solving this rational equation involves factoring the denominator of the first term, which is d2−2d−8d^2 - 2d - 8. Factoring this quadratic expression is essential for identifying the least common denominator (LCD) and simplifying the equation. The expression d2−2d−8d^2 - 2d - 8 can be factored into two binomials. We are looking for two numbers that multiply to -8 and add to -2. These numbers are -4 and 2. Therefore, we can rewrite the denominator as (d−4)(d+2)(d - 4)(d + 2). This factorization is a critical step because it reveals the factors that are present in the denominators of the other terms in the equation. By factoring the denominator, we gain a clearer picture of the structure of the equation and can proceed with finding the LCD. The factored form, (d−4)(d+2)(d - 4)(d + 2), also highlights the values of dd that would make the denominator zero, namely d=4d = 4 and d=−2d = -2. These values are crucial to identify as they represent restrictions on the possible solutions. Any solution that makes the denominator zero is an extraneous solution and must be excluded from the final answer. Factoring the denominator not only simplifies the equation but also provides valuable insights into the nature of the solutions and potential pitfalls. This step demonstrates the importance of mastering factoring techniques in solving rational equations. With the denominator factored, we can now rewrite the original equation, replacing d2−2d−8d^2 - 2d - 8 with its factored form, (d−4)(d+2)(d - 4)(d + 2). This sets the stage for the next step: finding the least common denominator.

Step 2: Identifying the Least Common Denominator (LCD)

After factoring the denominator d2−2d−8d^2 - 2d - 8 into (d−4)(d+2)(d - 4)(d + 2), the next crucial step is identifying the least common denominator (LCD). The LCD is the smallest expression that is divisible by each of the denominators in the equation. In our equation, −3d(d−4)(d+2)+3d−4=−2d+2\frac{-3 d}{(d-4)(d+2)}+\frac{3}{d-4}=\frac{-2}{d+2}, 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 consider each factor that appears in any of the denominators and include it in the LCD with the highest power to which it appears. In this case, the factors are (d−4)(d - 4) and (d+2)(d + 2). 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 factors: (d−4)(d+2)(d - 4)(d + 2). Understanding how to find the LCD is paramount in solving rational equations. The LCD allows us to eliminate the fractions by multiplying both sides of the equation by it. This transforms the rational equation into a simpler polynomial equation that is easier to solve. Furthermore, correctly identifying the LCD ensures that we are multiplying each term by the smallest possible expression, which minimizes the complexity of the resulting equation. By recognizing the LCD as (d−4)(d+2)(d - 4)(d + 2), we can now proceed with the next step, which involves multiplying both sides of the equation by the LCD to clear the fractions. This will lead us closer to isolating the variable and finding the solution.

Step 3: Multiplying by the LCD and Simplifying

With the LCD identified as (d−4)(d+2)(d - 4)(d + 2), the subsequent step involves multiplying both sides of the equation by the LCD and simplifying the resulting expression. This is a pivotal step in solving rational equations as it eliminates the fractions, transforming the equation into a more manageable form. Starting with the equation −3d(d−4)(d+2)+3d−4=−2d+2\frac{-3 d}{(d-4)(d+2)}+\frac{3}{d-4}=\frac{-2}{d+2}, we multiply each term on both sides by the LCD, (d−4)(d+2)(d - 4)(d + 2). This gives us:

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

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

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

This equation is now free of fractions, making it easier to solve. The next step is to distribute and combine like terms to further simplify the equation. Multiplying by the LCD is a crucial technique in solving rational equations, and this step demonstrates its effectiveness in transforming a complex equation into a simpler form. However, it's important to remember that this process can sometimes introduce extraneous solutions, which we will need to check later. For now, we proceed with simplifying the equation to isolate the variable.

Step 4: Distributing and Combining Like Terms

Following the multiplication by the LCD and simplification, the next step is to distribute and combine like terms in the equation −3d+3(d+2)=−2(d−4)-3d + 3(d + 2) = -2(d - 4). This process aims to further simplify the equation and bring it closer to a form where we can isolate the variable dd. First, we distribute the constants on both sides of the equation. On the left side, we distribute the 3 across (d+2)(d + 2), which gives us 3d+63d + 6. On the right side, we distribute the -2 across (d−4)(d - 4), which gives us −2d+8-2d + 8. The equation now looks like this:

−3d+3d+6=−2d+8-3d + 3d + 6 = -2d + 8

Next, we combine like terms on each side of the equation. On the left side, −3d-3d and +3d+3d cancel each other out, leaving us with just 6. The equation simplifies to:

6=−2d+86 = -2d + 8

This simplified equation is much easier to work with. We have eliminated the dd term on the left side, and now we need to isolate the dd term on the right side. Combining like terms is a fundamental algebraic technique that is essential in solving various types of equations. In this case, it has allowed us to significantly reduce the complexity of the equation, making it more straightforward to solve for the variable. With the equation simplified to this form, we can now proceed to isolate the variable dd by performing additional algebraic manipulations. This step sets the stage for finding the value(s) of dd that satisfy the equation.

Step 5: Isolating the Variable

After distributing and combining like terms, the equation is now 6=−2d+86 = -2d + 8. The crucial next step is isolating the variable dd. This involves performing algebraic operations to get dd by itself on one side of the equation. To begin, we want to eliminate the constant term on the right side of the equation, which is 8. We can do this by subtracting 8 from both sides of the equation:

6−8=−2d+8−86 - 8 = -2d + 8 - 8

This simplifies to:

−2=−2d-2 = -2d

Now, we have −2d-2d on the right side. To isolate dd, we need to divide both sides of the equation by the coefficient of dd, which is -2:

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

This simplifies to:

1=d1 = d

Therefore, we have found that d=1d = 1. Isolating the variable is a fundamental skill in algebra, and this step demonstrates its importance in solving equations. By performing inverse operations, we systematically eliminate terms and coefficients until the variable is by itself. In this case, we subtracted 8 from both sides and then divided both sides by -2 to isolate dd. Now that we have a potential solution, it's essential to remember the next critical step in solving rational equations: checking for extraneous solutions. This is particularly important because we multiplied both sides of the equation by an expression containing dd earlier in the process, which can sometimes introduce solutions that do not satisfy the original equation.

Step 6: Checking for Extraneous Solutions

Having found a potential solution, d=1d = 1, the crucial final step is checking for extraneous solutions. This step is particularly important when dealing with rational equations because multiplying both sides by an expression containing a variable can introduce solutions that do not satisfy the original equation. Extraneous solutions are values that satisfy the transformed equation but make one or more of the denominators in the original equation equal to zero, which is undefined. To check for extraneous solutions, we substitute the potential solution, d=1d = 1, back into the original equation: −3dd2−2d−8+3d−4=−2d+2\frac{-3 d}{d^2-2 d-8}+\frac{3}{d-4}=\frac{-2}{d+2}. Substituting d=1d = 1, we get:

−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}

Simplifying each term:

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

−3−9+3−3=−23\frac{-3}{-9}+\frac{3}{-3}=\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}

Since the equation holds true when d=1d = 1, this solution is valid and not extraneous. Checking for extraneous solutions is a critical step that ensures the accuracy of the solution. By substituting the potential solution back into the original equation, we verify that it does not lead to any undefined expressions or inconsistencies. In this case, d=1d = 1 satisfies the original equation, confirming that it is a valid solution. Therefore, the solution to the equation −3dd2−2d−8+3d−4=−2d+2\frac{-3 d}{d^2-2 d-8}+\frac{3}{d-4}=\frac{-2}{d+2} is indeed d=1d = 1.

Conclusion

In conclusion, the solution to the rational equation −3dd2−2d−8+3d−4=−2d+2\frac{-3 d}{d^2-2 d-8}+\frac{3}{d-4}=\frac{-2}{d+2} is d = 1. This solution was obtained through a systematic process involving factoring, identifying the least common denominator, multiplying to clear fractions, simplifying, isolating the variable, and, crucially, checking for extraneous solutions. Each step in the process is vital to arrive at the correct answer. Factoring the denominator d2−2d−8d^2 - 2d - 8 into (d−4)(d+2)(d - 4)(d + 2) allowed us to identify the LCD as (d−4)(d+2)(d - 4)(d + 2). Multiplying both sides of the equation by the LCD eliminated the fractions, transforming the equation into a simpler form. Simplifying the equation involved distributing, combining like terms, and isolating the variable dd. We found a potential solution of d=1d = 1. However, it was essential to check this solution in the original equation to ensure it was not extraneous. Substituting d=1d = 1 back into the original equation confirmed its validity. This step-by-step approach not only provides the solution but also highlights the key concepts and techniques involved in solving rational equations. Understanding these concepts is crucial for tackling similar problems in mathematics. The process underscores the importance of attention to detail, careful algebraic manipulation, and the critical step of verifying solutions to avoid extraneous results. By mastering these techniques, one can confidently solve a wide range of rational equations.