Adding Algebraic Fractions A Step-by-Step Guide To Solving X/(x^2-5x+6) + 3/(x+3)

In this comprehensive guide, we will meticulously dissect the process of adding two algebraic fractions: x/(x^2-5x+6) + 3/(x+3). This is a fundamental concept in algebra, and understanding it thoroughly is crucial for success in higher-level mathematics. We will break down each step, providing clear explanations and insights to ensure you grasp the underlying principles. This article aims to provide an in-depth exploration of algebraic fraction addition, focusing on a specific example to illustrate the general process. By understanding the steps involved in solving this problem, you'll be better equipped to tackle similar challenges and deepen your knowledge of algebraic manipulation.

Deconstructing the Problem

The problem we're tackling involves adding two fractions where the denominators are algebraic expressions. The key to successfully adding fractions lies in finding a common denominator. This allows us to combine the numerators and simplify the resulting expression. In the following sections, we will meticulously walk through each step, ensuring a clear understanding of the underlying principles and techniques involved in algebraic fraction addition. Our main goal is to illustrate how to combine these fractions into a single, simplified expression.

Step 1: Factoring the Denominator

Factoring the denominator is the crucial first step in simplifying the expression. We begin by focusing on the first fraction, x/(x^2-5x+6). The denominator, x^2-5x+6, is a quadratic expression that can be factored into two binomials. To do this, we look for two numbers that multiply to 6 (the constant term) and add up to -5 (the coefficient of the x term). These numbers are -2 and -3. Therefore, we can rewrite the denominator as (x-2)(x-3). This factorization is essential because it reveals the factors that will be crucial in finding a common denominator for the two fractions. The ability to factor quadratic expressions is a fundamental skill in algebra, and mastering it is crucial for success in simplifying and manipulating algebraic expressions. By rewriting the first fraction as x/((x-2)(x-3)), we gain a clearer understanding of its components and how it relates to the second fraction in the problem. This initial factorization sets the stage for the subsequent steps in solving the problem.

Thus, the expression becomes:

x/((x-2)(x-3)) + 3/(x+3)

This step is about identifying the building blocks of the denominator, setting the stage for finding a common denominator, which is the key to adding fractions.

Step 2: Finding the Common Denominator

Now that we've factored the denominator of the first fraction, the next crucial step is to find the common denominator. Looking at the two fractions, x/((x-2)(x-3)) and 3/(x+3), we can identify the different factors in their denominators: (x-2), (x-3), and (x+3). The common denominator must include all these factors. Therefore, the least common denominator (LCD) is the product of these unique factors: (x-2)(x-3)(x+3). To add the fractions, we need to rewrite each fraction with this common denominator. This involves multiplying the numerator and denominator of each fraction by the missing factors. For the first fraction, x/((x-2)(x-3)), we need to multiply both the numerator and the denominator by (x+3). For the second fraction, 3/(x+3), we need to multiply both the numerator and the denominator by (x-2)(x-3). This process ensures that we are adding equivalent fractions, maintaining the value of the original expression. By understanding how to find the common denominator, we can effectively combine fractions with different denominators into a single, simplified expression. This step is a fundamental concept in algebra and is essential for solving a wide range of problems involving fractions.

Applying this to our expression:

First fraction:

[x * (x+3)] / [(x-2)(x-3)(x+3)]

Second fraction:

[3 * (x-2)(x-3)] / [(x+3)(x-2)(x-3)]

This step transforms the fractions to have a common base for addition, which is a pivotal technique in fraction manipulation.

Step 3: Combining the Numerators

With a common denominator established, we can now focus on combining the numerators. This step involves adding the numerators of the fractions while keeping the common denominator. We have the expression: [x(x+3)] / [(x-2)(x-3)(x+3)] + [3(x-2)(x-3)] / [(x+3)(x-2)(x-3)]. First, we need to expand the numerators. For the first fraction, x(x+3) expands to x^2 + 3x. For the second fraction, 3(x-2)(x-3) requires a bit more work. We first multiply (x-2)(x-3), which gives us x^2 - 5x + 6. Then, we multiply this result by 3, yielding 3x^2 - 15x + 18. Now we can add the numerators: (x^2 + 3x) + (3x^2 - 15x + 18). Combining like terms, we get 4x^2 - 12x + 18. This resulting numerator is placed over the common denominator, giving us (4x^2 - 12x + 18) / [(x-2)(x-3)(x+3)]. The process of combining numerators is a fundamental operation in fraction addition, and it involves careful expansion and simplification of algebraic expressions. By accurately combining the numerators, we move closer to simplifying the overall expression and obtaining the final answer. This step highlights the importance of algebraic manipulation skills, such as expanding products and combining like terms, in solving complex problems.

Combining the numerators, we get:

(x(x+3) + 3(x-2)(x-3)) / ((x-2)(x-3)(x+3))

Expanding and simplifying the numerator:

(x^2 + 3x + 3(x^2 - 5x + 6)) / ((x-2)(x-3)(x+3))

(x^2 + 3x + 3x^2 - 15x + 18) / ((x-2)(x-3)(x+3))

(4x^2 - 12x + 18) / ((x-2)(x-3)(x+3))

This stage combines the individual fractions into a single fraction, ready for simplification, demonstrating the core mechanics of adding algebraic fractions.

Step 4: Simplifying the Expression

After combining the numerators, the next step is to simplify the resulting expression. We have (4x^2 - 12x + 18) / [(x-2)(x-3)(x+3)]. The first thing we should look for is any common factors in the numerator. We can factor out a 2 from the numerator: 2(2x^2 - 6x + 9). Now our expression is [2(2x^2 - 6x + 9)] / [(x-2)(x-3)(x+3)]. Next, we examine the quadratic expression in the numerator, 2x^2 - 6x + 9, to see if it can be factored further. However, in this case, it cannot be factored using integers. We also look for any common factors between the numerator and the denominator that can be canceled out. In this case, there are no common factors between 2(2x^2 - 6x + 9) and (x-2)(x-3)(x+3). Therefore, the simplified expression is [2(2x^2 - 6x + 9)] / [(x-2)(x-3)(x+3)]. Simplifying expressions is a crucial skill in algebra, as it allows us to present answers in their most concise and understandable form. This step often involves factoring, canceling common factors, and combining like terms. By simplifying the expression, we ensure that our answer is both accurate and easy to interpret. This final simplification completes the process of adding the two algebraic fractions.

Looking at the numerator, we can factor out a 2:

2(2x^2 - 6x + 9) / ((x-2)(x-3)(x+3))

The quadratic 2x^2 - 6x + 9 does not factor neatly, so we leave it as is. There are no common factors between the numerator and denominator, so the expression is simplified.

This final step is about presenting the answer in its most refined form, highlighting the importance of simplification in mathematical problem-solving.

Common Mistakes to Avoid

When working with algebraic fractions, there are several common mistakes to be aware of. One frequent error is failing to find a common denominator before adding the fractions. Remember, you cannot directly add fractions unless they have the same denominator. Another mistake is incorrectly factoring expressions, especially quadratic expressions. It's crucial to double-check your factoring to ensure accuracy. A third common mistake is forgetting to distribute when multiplying a factor across a numerator or denominator. For example, when multiplying 3(x-2)(x-3), make sure you correctly expand the expression. Finally, be careful when simplifying the final expression. Only cancel factors that are common to both the numerator and the denominator. Avoid the temptation to cancel terms that are part of a sum or difference. By being mindful of these common pitfalls, you can significantly reduce the likelihood of errors and improve your accuracy when working with algebraic fractions.

Conclusion: Mastering Algebraic Fraction Addition

In conclusion, mastering the addition of algebraic fractions requires a systematic approach and a solid understanding of algebraic principles. The steps outlined in this guide – factoring, finding a common denominator, combining numerators, and simplifying – provide a clear roadmap for tackling these types of problems. By practicing these steps and being mindful of common mistakes, you can build confidence and proficiency in working with algebraic fractions. This skill is not only essential for success in algebra but also forms a foundation for more advanced mathematical concepts. Remember, the key to success is consistent practice and a willingness to break down complex problems into manageable steps.