Solving The Equation 2/(d-2) = (3d)/(4d+12) A Comprehensive Guide

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In this article, we will delve into the process of solving the algebraic equation 2/(d-2) = (3d)/(4d+12). This equation involves fractions with variables in the denominators, requiring careful manipulation to isolate the variable d. We will explore the steps involved, from identifying restrictions on the variable to applying algebraic techniques such as cross-multiplication and solving quadratic equations. By the end of this comprehensive explanation, you will have a clear understanding of how to solve this type of equation and similar algebraic problems. This problem falls under the domain of mathematics, specifically algebra, which deals with symbols and the rules for manipulating those symbols. Equations like this are fundamental in various fields, including physics, engineering, and economics, where mathematical models are used to represent real-world situations. Therefore, mastering the techniques to solve such equations is crucial for anyone pursuing studies or careers in these areas. Throughout our discussion, we will emphasize the importance of checking for extraneous solutions, which can arise when dealing with rational equations. These are solutions that satisfy the transformed equation but not the original equation due to restrictions on the variable. We'll also highlight the significance of understanding the underlying principles of algebra, such as the properties of equality and the order of operations, which are essential for accurate and efficient problem-solving. Before we dive into the solution, let's first understand the different components of the equation and the concepts involved. The equation 2/(d-2) = (3d)/(4d+12) is a rational equation because it involves rational expressions, which are fractions where the numerator and denominator are polynomials. Solving rational equations requires a slightly different approach compared to solving linear equations or simple polynomial equations. It is essential to identify the values of the variable that would make the denominators zero, as these values are not allowed and must be excluded from the solution set. By systematically working through the steps, we'll arrive at the solution and gain a deeper understanding of the principles of algebraic problem-solving.

1. Identifying Restrictions

Before we begin solving the equation 2/(d-2) = (3d)/(4d+12), it is crucial to identify any restrictions on the variable d. Restrictions occur when the denominator of a fraction equals zero, as division by zero is undefined. This step is vital to avoid extraneous solutions, which are solutions that emerge during the solving process but do not satisfy the original equation. In the given equation, we have two denominators: (d-2) and (4d+12). We need to determine the values of d that would make each of these denominators equal to zero. Let's start with the first denominator, (d-2). To find the restriction, we set (d-2) equal to zero and solve for d:

  • d - 2 = 0
  • d = 2

So, d cannot be equal to 2, as this would make the first denominator zero. Now, let's consider the second denominator, (4d+12). We set (4d+12) equal to zero and solve for d:

  • 4d + 12 = 0
  • 4d = -12
  • d = -12 / 4
  • d = -3

Therefore, d cannot be equal to -3, as this would make the second denominator zero. In summary, we have identified two restrictions on the variable d: d ≠ 2 and d ≠ -3. These restrictions mean that if we obtain solutions of d = 2 or d = -3 during the solving process, we must discard them as extraneous solutions. Keeping track of these restrictions is a critical step in solving rational equations. Now that we have established the restrictions, we can proceed with the algebraic manipulations to solve the equation. Remember, the goal is to isolate the variable d while adhering to the rules of algebra. The next step typically involves cross-multiplication, which allows us to eliminate the fractions and transform the equation into a more manageable form. However, before we do that, it's always a good practice to simplify the equation if possible. In this case, we can notice that the denominator (4d+12) can be factored, which might lead to some simplification. We will explore this in the next section.

2. Simplifying the Equation

Before we proceed with cross-multiplication in the equation 2/(d-2) = (3d)/(4d+12), it's often beneficial to simplify the equation if possible. Simplification can make the subsequent steps easier and reduce the chances of errors. In this case, we can observe that the denominator (4d+12) on the right side of the equation has a common factor. Factoring out the greatest common factor (GCF) can lead to a simpler form. The GCF of (4d+12) is 4. We can factor out 4 from the expression:

  • 4d + 12 = 4(d + 3)

Now, let's substitute this factored form back into the original equation:

  • 2/(d-2) = (3d) / [4(d+3)]

With the factored form, the equation now looks like this: 2/(d-2) = (3d) / [4(d+3)]. This simplified form can make cross-multiplication less cumbersome. The next step is to apply cross-multiplication to eliminate the fractions. Cross-multiplication involves multiplying the numerator of the left fraction by the denominator of the right fraction and vice versa. This process transforms the equation into a linear or polynomial equation, which is generally easier to solve. In our case, cross-multiplying the equation 2/(d-2) = (3d) / [4(d+3)] will give us a quadratic equation. Quadratic equations are polynomial equations of the second degree, and they can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. It's important to choose the method that is most appropriate for the given equation. Once we have obtained the solutions, we must always remember to check them against the restrictions we identified earlier. If any of the solutions violate the restrictions, they are extraneous and must be discarded. By simplifying the equation before cross-multiplication, we have potentially reduced the complexity of the resulting quadratic equation. This can save time and effort in the long run. Now, let's proceed with cross-multiplication and see what quadratic equation we obtain. This step will bring us closer to solving for the value of d. Remember, accuracy and careful manipulation are key to successfully solving algebraic equations.

3. Cross-Multiplication

Having simplified the equation to 2/(d-2) = (3d) / [4(d+3)], the next step is to perform cross-multiplication. Cross-multiplication is a technique used to eliminate fractions in an equation by multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. In this case, we will multiply 2 by (4(d+3)) and (3d) by (d-2). This gives us the following equation:

  • 2 * [4(d+3)] = 3d * (d-2)

Now, let's simplify both sides of the equation. First, we distribute the 2 on the left side:

  • 8(d+3) = 3d(d-2)

Next, we distribute the 8 on the left side and the (3d) on the right side:

  • 8d + 24 = 3d^2 - 6d

Now we have a quadratic equation. To solve a quadratic equation, we generally want to set it equal to zero. So, we will move all the terms to one side of the equation. Let's subtract (8d) and 24 from both sides:

  • 0 = 3d^2 - 6d - 8d - 24

Combine like terms:

  • 0 = 3d^2 - 14d - 24

Now we have the quadratic equation 3d^2 - 14d - 24 = 0. This equation is in the standard form of a quadratic equation, which is ax^2 + bx + c = 0, where a, b, and c are constants. In our case, a = 3, b = -14, and c = -24. To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Factoring is often the quickest method if the quadratic equation can be factored easily. However, if factoring is not straightforward, the quadratic formula is a reliable alternative. The quadratic formula is given by:

  • d = [-b ± √(b^2 - 4ac)] / (2a)

In the next section, we will explore how to solve this quadratic equation, either by factoring or using the quadratic formula. Once we have the solutions, we will need to check them against the restrictions we identified earlier to ensure they are not extraneous. Remember, the goal is to find the values of d that satisfy the original equation while adhering to the rules of algebra. Careful manipulation and attention to detail are essential for success.

4. Solving the Quadratic Equation

We have arrived at the quadratic equation 3d^2 - 14d - 24 = 0. To solve this equation, we can either attempt to factor it or use the quadratic formula. Let's first try factoring. Factoring involves finding two binomials that, when multiplied together, give us the quadratic expression. We are looking for two binomials of the form (Ad + B)(Cd + D) such that:

  • AC = 3
  • AD + BC = -14
  • BD = -24

After some trial and error, we can find that the quadratic expression can be factored as:

  • (3d + 4)(d - 6) = 0

Now, to find the solutions for d, we set each factor equal to zero and solve:

  • 3d + 4 = 0 or d - 6 = 0

Solving the first equation:

  • 3d = -4
  • d = -4/3

Solving the second equation:

  • d = 6

So, we have obtained two potential solutions: d = -4/3 and d = 6. Now, before we declare these as the solutions, we must check them against the restrictions we identified at the beginning of the problem. Recall that the restrictions were d ≠ 2 and d ≠ -3. Our potential solutions, d = -4/3 and d = 6, do not violate these restrictions. Therefore, both solutions are valid. If we had trouble factoring the quadratic equation, we could have used the quadratic formula instead. The quadratic formula is a general method for solving quadratic equations of the form ax^2 + bx + c = 0, and it is given by:

  • d = [-b ± √(b^2 - 4ac)] / (2a)

In our case, a = 3, b = -14, and c = -24. Plugging these values into the quadratic formula, we would get the same solutions: d = -4/3 and d = 6. The choice of method, whether factoring or using the quadratic formula, often depends on the specific equation and personal preference. Factoring can be quicker if the quadratic expression is easily factorable, while the quadratic formula is a reliable method that always works. In the next section, we will present our final solutions and summarize the steps we took to solve the equation.

5. Checking for Extraneous Solutions and Final Solutions

We have found two potential solutions for the equation 2/(d-2) = (3d)/(4d+12): d = -4/3 and d = 6. However, before we finalize these solutions, it's crucial to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation (in this case, the quadratic equation) but do not satisfy the original equation. They typically arise when we perform operations that are not reversible, such as squaring both sides or, in this case, multiplying both sides by expressions that contain the variable. To check for extraneous solutions, we need to substitute each potential solution back into the original equation and see if it holds true. Let's start with d = -4/3. Substituting this value into the original equation:

  • 2/((-4/3)-2) = (3*(-4/3))/(4*(-4/3)+12)

Simplify the denominators:

  • 2/((-4/3)-(6/3)) = (-4)/((-16/3)+(36/3))
  • 2/(-10/3) = (-4)/(20/3)

Now, divide the fractions:

  • 2 * (-3/10) = (-4) * (3/20)
  • -6/10 = -12/20
  • -3/5 = -3/5

Since the equation holds true, d = -4/3 is a valid solution. Now, let's check the second potential solution, d = 6. Substituting this value into the original equation:

  • 2/(6-2) = (36)/(46+12)
  • 2/4 = 18/(24+12)
  • 2/4 = 18/36
  • 1/2 = 1/2

Since the equation also holds true for d = 6, this is also a valid solution. We have checked both potential solutions against the original equation and confirmed that neither of them is extraneous. Additionally, we identified at the beginning that d cannot be 2 or -3. Neither of our solutions equals these values, thus confirming they are valid solutions. Therefore, the solutions to the equation 2/(d-2) = (3d)/(4d+12) are d = -4/3 and d = 6. In conclusion, solving equations involving rational expressions requires careful attention to detail, especially when dealing with restrictions and extraneous solutions. By following a systematic approach, we can successfully find the solutions and gain a deeper understanding of algebraic principles. This process involved identifying restrictions, simplifying the equation, cross-multiplication, solving the resulting quadratic equation, and finally, checking for extraneous solutions. This comprehensive approach is essential for accuracy and success in solving algebraic problems.

In this comprehensive exploration, we successfully solved the equation 2/(d-2) = (3d)/(4d+12), demonstrating the critical steps involved in handling rational equations. We began by emphasizing the importance of identifying restrictions on the variable d to avoid extraneous solutions, recognizing that division by zero is undefined. This led us to the restrictions d ≠ 2 and d ≠ -3. Subsequently, we simplified the equation by factoring the denominator (4d+12), making the equation easier to manipulate. This simplification step is a valuable technique in problem-solving, as it often reduces the complexity of subsequent calculations. Following simplification, we employed cross-multiplication to eliminate the fractions, transforming the equation into a quadratic form: 3d^2 - 14d - 24 = 0. This transformation is a key step in solving rational equations, as it allows us to apply familiar methods for solving quadratic equations. We then tackled the quadratic equation by factoring, successfully expressing it as (3d + 4)(d - 6) = 0. This factorization led us to two potential solutions: d = -4/3 and d = 6. However, obtaining potential solutions is not the end of the process. We stressed the necessity of checking these solutions against the original equation to ensure they are valid. This check is particularly important in rational equations, where extraneous solutions can arise due to the initial restrictions. By substituting each potential solution back into the original equation, we confirmed that both d = -4/3 and d = 6 satisfy the equation and do not violate any restrictions. Therefore, we confidently declared these as the final solutions. This process highlights the meticulous nature of mathematical problem-solving. It's not enough to simply find potential solutions; verifying their validity is crucial for accuracy. This article has not only provided a step-by-step solution to the given equation but also emphasized the underlying principles and techniques applicable to a broader range of algebraic problems. Understanding these principles, such as the properties of equality, the importance of restrictions, and the methods for solving quadratic equations, is essential for success in mathematics and related fields. In summary, solving rational equations like 2/(d-2) = (3d)/(4d+12) involves a systematic approach: identifying restrictions, simplifying the equation, eliminating fractions, solving the resulting equation, and verifying the solutions. This comprehensive methodology ensures accuracy and a deeper understanding of the mathematical concepts involved.