Solving 2/(d-2) = (3d)/(4d+12) A Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the intricacies of solving the algebraic equation 2/(d-2) = 3d/(4d+12). This equation, which falls under the realm of rational equations, demands a methodical approach to ensure accurate solutions. Rational equations are equations that contain fractions with polynomials in the numerator and/or the denominator. The challenge in solving these equations lies in handling the denominators and avoiding extraneous solutions. We will dissect the problem step-by-step, covering essential techniques such as cross-multiplication, simplification, and solving quadratic equations. Our goal is to provide a clear, easy-to-follow explanation that empowers you to tackle similar algebraic challenges with confidence. We will explore the conditions under which solutions are valid, emphasizing the importance of checking for extraneous solutions, which can arise when dealing with rational expressions. The concepts discussed here are fundamental in algebra and have wide applications in various fields of mathematics and beyond. This detailed exploration will not only equip you with the tools to solve this specific equation but also enhance your overall problem-solving skills in algebra. The step-by-step approach will break down the complexity, making it accessible to learners of all levels. We will also highlight common pitfalls and strategies to avoid them, ensuring a thorough understanding of the solution process.

Step 1: Identifying Restrictions

Before we dive into the algebraic manipulation, it's crucial to identify the restrictions on the variable d. These restrictions stem from the denominators of the fractions in the equation. A fundamental principle in mathematics is that division by zero is undefined. Therefore, any value of d that makes the denominator of either fraction equal to zero must be excluded from the solution set. In the given equation, 2/(d-2) = 3d/(4d+12), we have two denominators to consider: (d-2) and (4d+12). Setting each of these equal to zero will help us find the restricted values. For the first denominator, we have d - 2 = 0. Adding 2 to both sides gives us d = 2. This means that d cannot be equal to 2, as it would make the first denominator zero. For the second denominator, we have 4d + 12 = 0. Subtracting 12 from both sides gives 4d = -12. Dividing both sides by 4 yields d = -3. Hence, d cannot be equal to -3, as it would make the second denominator zero. Thus, the restrictions on d are d ≠ 2 and d ≠ -3. These restrictions are paramount because any solution we find later must be checked against these values. If a solution coincides with a restricted value, it is considered an extraneous solution and must be discarded. Ignoring these restrictions can lead to incorrect answers, underscoring the importance of this initial step. Recognizing and noting these restrictions is a cornerstone of solving rational equations accurately. By understanding these constraints, we set the stage for a rigorous and valid solution process. This initial analysis ensures that our final solutions are mathematically sound and consistent with the original equation.

Step 2: Cross-Multiplication

The next step in solving the equation 2/(d-2) = 3d/(4d+12) is to eliminate the fractions. A common technique for doing this is cross-multiplication. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This method effectively clears the denominators and transforms the equation into a more manageable form. Applying cross-multiplication to our equation, we multiply 2 by (4d + 12) and 3d by (d - 2). This gives us the equation: 2(4d + 12) = 3d(d - 2). Now, we have an equation without fractions, which simplifies the subsequent steps. It's crucial to perform this step accurately, as any error here will propagate through the rest of the solution. The resulting equation is a crucial intermediate stage that allows us to proceed with algebraic simplification. This transformation is a standard technique for handling rational equations, making the problem more accessible. By eliminating the fractions, we set the stage for expanding, simplifying, and ultimately solving for d. Cross-multiplication is a powerful tool in algebra, and mastering its application is essential for solving a wide range of equations. It's a technique that converts a complex rational equation into a simpler polynomial equation, making it easier to find the solution. The resulting equation from cross-multiplication is a significant milestone in the solution process, as it sets the foundation for the next steps in solving for the unknown variable.

Step 3: Expanding and Simplifying

With the fractions eliminated through cross-multiplication, the next crucial step is to expand and simplify the resulting equation. From the previous step, we have 2(4d + 12) = 3d(d - 2). To expand this equation, we distribute the constants and variables on both sides. On the left side, we multiply 2 by both 4d and 12, which yields 8d + 24. On the right side, we distribute 3d across (d - 2), resulting in 3d² - 6d. So, our expanded equation is 8d + 24 = 3d² - 6d. Now, to simplify this equation further, we need to rearrange it into a standard form, typically by setting one side equal to zero. This often involves combining like terms and moving all terms to one side of the equation. A standard approach is to move all terms to the side with the highest degree term, which in this case is 3d². Subtracting 8d and 24 from both sides, we get 0 = 3d² - 6d - 8d - 24. Combining the like terms, specifically the d terms, we have -6d - 8d = -14d. Thus, the simplified equation is 0 = 3d² - 14d - 24. This quadratic equation is now in standard form, ax² + bx + c = 0, where a = 3, b = -14, and c = -24. Having the equation in this form is essential for solving it using various methods, such as factoring, completing the square, or applying the quadratic formula. The process of expanding and simplifying is a fundamental algebraic skill, and it is crucial for solving not just this equation but a wide range of mathematical problems. Accuracy in these steps is vital, as any error can lead to an incorrect solution. By meticulously expanding and simplifying, we transform the equation into a manageable form that is ready for the next stage of solving.

Step 4: Solving the Quadratic Equation

Now that we have the simplified quadratic equation 3d² - 14d - 24 = 0, the next step is to solve for d. There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, let's explore factoring and the quadratic formula. First, we attempt to factor the quadratic equation. Factoring involves expressing the quadratic expression as a product of two binomials. We look for two numbers that multiply to give the product of the leading coefficient (3) and the constant term (-24), which is -72, and add up to the middle coefficient (-14). The numbers -18 and 4 satisfy these conditions since (-18) * 4 = -72 and -18 + 4 = -14. We can rewrite the middle term using these numbers: 3d² - 18d + 4d - 24 = 0. Now, we factor by grouping: 3d(d - 6) + 4(d - 6) = 0. We can factor out the common binomial (d - 6), giving us (3d + 4)(d - 6) = 0. Setting each factor equal to zero gives the solutions: 3d + 4 = 0 which leads to d = -4/3, and d - 6 = 0 which leads to d = 6. Alternatively, if factoring is not straightforward, we can use the quadratic formula, which is given by: d = [-b ± √(b² - 4ac)] / (2a). For our equation, a = 3, b = -14, and c = -24. Plugging these values into the formula, we get: d = [14 ± √((-14)² - 4 * 3 * -24)] / (2 * 3). Simplifying, we have: d = [14 ± √(196 + 288)] / 6, which further simplifies to d = [14 ± √484] / 6. Since √484 = 22, we have: d = [14 ± 22] / 6. This gives us two solutions: d = (14 + 22) / 6 = 36 / 6 = 6, and d = (14 - 22) / 6 = -8 / 6 = -4/3. Both factoring and the quadratic formula yield the same solutions, which are d = 6 and d = -4/3. These values are potential solutions to the original equation, but we must check them against the restrictions we identified earlier.

Step 5: Checking for Extraneous Solutions

After finding the potential solutions for d, it's imperative to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but do not satisfy the original equation. These solutions often arise when dealing with rational equations because certain values of the variable can make the denominator of a fraction equal to zero, which is undefined. Recall that in Step 1, we identified the restrictions on d as d ≠ 2 and d ≠ -3. These are the values that would make the denominators of the original equation zero. We found two potential solutions: d = 6 and d = -4/3. Now, we need to check if these values violate our restrictions. First, let's consider d = 6. Since 6 is not equal to 2 or -3, it does not violate any restrictions. We can substitute d = 6 back into the original equation to verify if it is a valid solution: 2 / (6 - 2) = 3 * 6 / (4 * 6 + 12). This simplifies to 2 / 4 = 18 / (24 + 12), which further simplifies to 1/2 = 18 / 36, and finally 1/2 = 1/2. Since the equation holds true, d = 6 is a valid solution. Next, let's consider d = -4/3. Again, since -4/3 is not equal to 2 or -3, it does not violate any restrictions. We substitute d = -4/3 back into the original equation: 2 / (-4/3 - 2) = 3 * (-4/3) / (4 * (-4/3) + 12). Simplifying the left side, we get 2 / (-4/3 - 6/3) = 2 / (-10/3) = -6/10 = -3/5. Simplifying the right side, we get (-4) / (-16/3 + 36/3) = -4 / (20/3) = -12/20 = -3/5. Since both sides of the equation are equal, d = -4/3 is also a valid solution. Therefore, neither of our potential solutions is extraneous, and both d = 6 and d = -4/3 are valid solutions to the original equation. Checking for extraneous solutions is a critical step in solving rational equations, ensuring that the solutions we find are mathematically sound and consistent with the original problem.

Conclusion

In conclusion, we have successfully solved the equation 2/(d-2) = 3d/(4d+12) using a step-by-step approach. We began by identifying the restrictions on d to avoid division by zero, finding that d ≠ 2 and d ≠ -3. Next, we used cross-multiplication to eliminate the fractions, resulting in the equation 2(4d + 12) = 3d(d - 2). We then expanded and simplified this equation to obtain the quadratic equation 3d² - 14d - 24 = 0. To solve the quadratic equation, we employed both factoring and the quadratic formula, which led us to the potential solutions d = 6 and d = -4/3. Finally, we meticulously checked these solutions against the restrictions we identified earlier. Both values were found to be valid, meaning they did not violate the restrictions and satisfied the original equation. Therefore, the solutions to the equation 2/(d-2) = 3d/(4d+12) are d = 6 and d = -4/3. This comprehensive process highlights the importance of each step in solving rational equations, from identifying restrictions to verifying solutions. By following these steps carefully, we can accurately solve complex algebraic equations and avoid extraneous solutions. The techniques discussed here are fundamental in algebra and can be applied to a wide range of problems. Understanding these concepts and practicing these skills will significantly enhance your problem-solving abilities in mathematics. This detailed exploration provides a solid foundation for tackling similar algebraic challenges with confidence and precision.