Step-by-Step Guide Simplifying Rational Expression (25/(x^2+9-6x)) - (5/6)

In the realm of algebra, simplifying expressions is a fundamental skill. This article will guide you through the process of simplifying a specific rational expression. We'll break down each step involved in combining two fractions with different denominators into a single, simplified fraction. This comprehensive guide aims to provide a clear understanding of the techniques used in simplifying rational expressions, ensuring you can confidently tackle similar problems.

Understanding the Problem

The problem at hand involves the difference of two rational expressions: (25/(x^2+9-6x)) - (5/6). Our goal is to combine these two expressions into a single, simplified rational expression. This requires finding a common denominator, performing the subtraction, and then simplifying the resulting expression by factoring and canceling common factors. This initial step is crucial as it sets the stage for the subsequent algebraic manipulations. A clear understanding of the initial problem allows for a more focused and efficient approach to the simplification process. Before diving into the solution, it's important to recognize the structure of the given expression. We have two fractions, and our objective is to merge them into one simplified fraction. This process typically involves finding a common denominator, adjusting the numerators accordingly, and then combining them. This preliminary assessment helps in formulating a strategy for simplification. The complexity arises from the fact that the denominators are different, one being a quadratic expression and the other a constant. This difference necessitates a careful approach to finding the least common denominator (LCD), which is the foundation for combining the fractions. The first denominator, x^2 + 9 - 6x, is a quadratic expression, hinting at the possibility of factoring it. Factoring is a powerful tool in simplifying rational expressions because it allows us to identify common factors that can be canceled out later. The second denominator is simply 6, which is a constant. This contrast in the nature of the denominators suggests that we need to pay close attention to the algebraic manipulations involved in finding the LCD. In essence, our task is to transform the given expression into a simpler, more manageable form. This involves a series of algebraic steps, each building upon the previous one. The ultimate goal is to arrive at an expression that is both equivalent to the original and in its simplest form. This simplified form often reveals underlying structures and relationships that might not be immediately apparent in the original expression. Understanding the problem thoroughly is the first step towards successfully navigating the simplification process.

Step 1: Factoring the Denominator

The first critical step in simplifying the expression (25/(x^2+9-6x)) - (5/6) is to factor the quadratic denominator. The denominator x^2 + 9 - 6x can be rearranged as x^2 - 6x + 9. This quadratic expression is a perfect square trinomial, which can be factored into (x - 3)^2. Factoring the denominator is essential because it allows us to identify common factors and find the least common denominator (LCD) more easily. By expressing the denominator in its factored form, we can better understand its structure and how it relates to other expressions in the problem. This step is not just about applying a factoring technique; it's about revealing the underlying algebraic structure of the expression. The factored form (x - 3)^2 provides valuable information about the expression's behavior and its potential for simplification. A perfect square trinomial is a special type of quadratic expression that follows a specific pattern: a^2 - 2ab + b^2 = (a - b)^2 or a^2 + 2ab + b^2 = (a + b)^2. Recognizing this pattern is crucial for efficient factoring. In our case, x^2 - 6x + 9 fits the pattern of a^2 - 2ab + b^2, where a = x and b = 3. Applying this pattern, we can directly factor the expression as (x - 3)^2. The factored form (x - 3)^2 tells us that the expression has a repeated root at x = 3. This information can be useful in various contexts, such as solving equations or analyzing the behavior of the expression. However, in the context of simplifying rational expressions, the primary benefit of factoring is to identify common factors that can be canceled out. Factoring the denominator is a key step in simplifying rational expressions because it often leads to significant simplifications. By expressing the denominator in its factored form, we can more easily identify common factors with the numerator or other denominators. This, in turn, allows us to cancel out these factors, resulting in a simpler expression. In this specific case, factoring the denominator as (x - 3)^2 sets the stage for finding the least common denominator and combining the two fractions. It's a crucial step that lays the foundation for the subsequent algebraic manipulations. Factoring is not just a mechanical process; it's a way of gaining deeper insight into the structure of an expression. By factoring, we are essentially breaking down the expression into its fundamental components, which can reveal hidden relationships and simplifications. This understanding is essential for mastering algebraic manipulations and solving more complex problems.

Step 2: Finding the Least Common Denominator (LCD)

Once we've factored the denominator, the next step is to find the least common denominator (LCD). The given expression is now (25/(x-3)^2) - (5/6). The LCD is the smallest expression that is divisible by both denominators. In this case, the denominators are (x - 3)^2 and 6. The LCD is the product of the unique factors, each raised to the highest power that appears in either denominator. Therefore, the LCD is 6(x - 3)^2. Finding the LCD is a fundamental step in adding or subtracting fractions, whether they are numerical or algebraic. It allows us to rewrite the fractions with a common denominator, making it possible to combine them. The LCD ensures that we are working with the smallest possible common denominator, which simplifies the subsequent calculations. In the context of rational expressions, the LCD can be more complex than in numerical fractions. It may involve variables and exponents, as in this case. However, the underlying principle remains the same: to find the smallest expression that is divisible by all the denominators. The process of finding the LCD involves identifying the unique factors in each denominator and then taking the highest power of each factor. In our case, the factors are (x - 3) and 6. The highest power of (x - 3) is 2, and the highest power of 6 is 1. Therefore, the LCD is 6(x - 3)^2. It's important to note that the LCD must include all the factors from each denominator. If a factor appears in more than one denominator, we take the highest power of that factor. This ensures that the LCD is divisible by all the denominators. Finding the LCD is not just a mechanical process; it requires a careful analysis of the denominators and their factors. A clear understanding of the concept of LCD is essential for successfully simplifying rational expressions. The LCD serves as the bridge that allows us to combine the fractions. Once we have the LCD, we can rewrite each fraction with this common denominator, making it possible to add or subtract the numerators. This step is crucial for simplifying the expression and arriving at a single, simplified fraction. Without the LCD, we cannot directly combine the fractions. The LCD provides the necessary framework for performing the subtraction and simplifying the expression. It's a fundamental concept that underpins the entire process of simplifying rational expressions. Understanding the LCD is also essential for solving equations involving fractions. When we have an equation with fractions, we can multiply both sides of the equation by the LCD to eliminate the fractions. This simplifies the equation and makes it easier to solve. Therefore, mastering the concept of LCD is crucial for a wide range of algebraic manipulations.

Step 3: Rewriting Fractions with the LCD

With the LCD determined as 6(x - 3)^2, we must now rewrite each fraction with this common denominator. This involves multiplying the numerator and denominator of each fraction by the appropriate factors. For the first fraction, (25/(x - 3)^2), we need to multiply both the numerator and denominator by 6. This gives us (25 * 6) / (6(x - 3)^2) = 150 / (6(x - 3)^2). For the second fraction, (5/6), we need to multiply both the numerator and denominator by (x - 3)^2. This gives us (5(x - 3)^2) / (6(x - 3)^2). Rewriting fractions with a common denominator is a crucial step in adding or subtracting fractions. It ensures that we are working with equivalent fractions that can be directly combined. The key is to multiply both the numerator and denominator by the same factor, which doesn't change the value of the fraction. In the context of rational expressions, this step may involve multiplying by algebraic expressions, as in this case. However, the underlying principle remains the same: to create equivalent fractions with a common denominator. The process of rewriting fractions with the LCD involves careful attention to the factors needed to achieve the common denominator. We need to identify what factors are missing in each denominator and then multiply both the numerator and denominator by those factors. This ensures that we are creating equivalent fractions that can be combined. It's important to note that we are not changing the value of the fraction; we are simply changing its form. By multiplying both the numerator and denominator by the same factor, we are essentially multiplying the fraction by 1, which doesn't alter its value. This is a fundamental principle in working with fractions. Rewriting fractions with the LCD is a prerequisite for adding or subtracting them. Once we have the fractions in this form, we can simply add or subtract the numerators and keep the common denominator. This simplifies the process of combining the fractions and arriving at a single fraction. Without rewriting the fractions with the LCD, we cannot directly add or subtract them. The LCD provides the necessary framework for performing the operation and simplifying the expression. It's a fundamental concept that underpins the entire process of adding or subtracting fractions. Rewriting fractions with the LCD is also essential for solving equations involving fractions. When we have an equation with fractions, we can rewrite them with the LCD and then combine them into a single fraction. This simplifies the equation and makes it easier to solve. Therefore, mastering this step is crucial for a wide range of algebraic manipulations. This step highlights the importance of understanding equivalent fractions and how to manipulate them. By rewriting fractions with the LCD, we are demonstrating our understanding of these concepts and our ability to apply them in algebraic contexts.

Step 4: Combining the Numerators

Now that both fractions have the common denominator 6(x - 3)^2, we can combine the numerators. The expression becomes (150 - 5(x - 3)^2) / (6(x - 3)^2). This step involves subtracting the second numerator from the first, while keeping the common denominator. Combining the numerators is a direct application of the rules of fraction arithmetic. Once we have a common denominator, we can simply add or subtract the numerators and keep the denominator the same. This step is crucial for simplifying the expression and arriving at a single fraction. In the context of rational expressions, the numerators may be algebraic expressions, as in this case. This means that we need to be careful with the order of operations and the signs when combining the numerators. However, the underlying principle remains the same: to add or subtract the numerators and keep the common denominator. The process of combining the numerators involves paying close attention to the signs and the order of operations. We need to distribute any negative signs and combine like terms. This may involve expanding algebraic expressions and simplifying the resulting expression. It's important to be meticulous and avoid making careless errors. The combined numerator represents the difference between the two original numerators, expressed in terms of the common denominator. This expression is crucial for the subsequent simplification steps. It encapsulates the result of the subtraction and sets the stage for factoring and canceling common factors. Combining the numerators is a key step in simplifying rational expressions because it brings us closer to a single, simplified fraction. By combining the numerators, we are reducing the complexity of the expression and making it easier to manipulate. This step is a direct application of the rules of fraction arithmetic and highlights the importance of understanding these rules. The combined numerator is not necessarily in its simplest form. It may contain common factors that can be canceled out with the denominator. Therefore, the next step in the simplification process is to factor the numerator and look for common factors. Combining the numerators is a fundamental step in simplifying rational expressions and is a prerequisite for the subsequent steps. It's a direct application of the rules of fraction arithmetic and highlights the importance of understanding these rules. This step demonstrates our ability to apply algebraic principles in a systematic and organized manner. By combining the numerators, we are moving closer to the final simplified expression.

Step 5: Expanding and Simplifying the Numerator

Before we can factor, we need to expand and simplify the numerator. Starting with 150 - 5(x - 3)^2, we first expand (x - 3)^2 to get x^2 - 6x + 9. Then, we multiply this by -5, resulting in -5x^2 + 30x - 45. Finally, we combine this with 150 to get -5x^2 + 30x + 105. Expanding and simplifying the numerator is a crucial step in simplifying rational expressions. It allows us to combine like terms and put the numerator in a form that is easier to factor. This step may involve distributing, multiplying, and combining terms. The goal is to transform the numerator into a simpler, more manageable expression. In the context of rational expressions, the numerator may be a complex algebraic expression, as in this case. This means that we need to be careful with the order of operations and the signs when expanding and simplifying. However, the underlying principle remains the same: to transform the numerator into a simpler form. The process of expanding and simplifying the numerator involves careful attention to the order of operations and the rules of algebra. We need to distribute any factors, multiply expressions, and combine like terms. This may require multiple steps and careful attention to detail. The simplified numerator represents the result of the algebraic manipulations performed on the original numerator. This expression is crucial for the subsequent factoring step. It encapsulates the simplified form of the numerator and sets the stage for identifying common factors with the denominator. Expanding and simplifying the numerator is a key step in simplifying rational expressions because it makes the numerator easier to factor. A factored numerator allows us to identify common factors with the denominator, which can then be canceled out. This is a crucial step in reducing the expression to its simplest form. The simplified numerator is not necessarily in its final form. It may still contain common factors that can be factored out. Therefore, the next step in the simplification process is to factor the numerator and look for common factors. Expanding and simplifying the numerator is a fundamental step in simplifying rational expressions and is a prerequisite for the subsequent factoring step. It's a direct application of the rules of algebra and highlights the importance of understanding these rules. This step demonstrates our ability to apply algebraic principles in a systematic and organized manner. By expanding and simplifying the numerator, we are moving closer to the final simplified expression. This process often involves recognizing patterns and applying algebraic identities, such as the square of a binomial. In this case, recognizing that (x - 3)^2 is the square of a binomial allows us to efficiently expand it.

Step 6: Factoring the Numerator

Now that we have the simplified numerator, -5x^2 + 30x + 105, we need to factor it. First, we can factor out a common factor of -5, resulting in -5(x^2 - 6x - 21). Factoring the numerator is a crucial step in simplifying rational expressions. It allows us to identify common factors with the denominator, which can then be canceled out. This step is essential for reducing the expression to its simplest form. In the context of rational expressions, the numerator may be a complex algebraic expression, as in this case. This means that we need to be familiar with various factoring techniques, such as factoring out a common factor, factoring quadratic expressions, and using special factoring patterns. However, the underlying principle remains the same: to express the numerator as a product of factors. The process of factoring the numerator involves careful analysis of the expression and the application of appropriate factoring techniques. We need to identify common factors, look for patterns, and try different combinations until we find the factors. This may require multiple steps and careful attention to detail. The factored numerator represents the expression as a product of factors. This form is crucial for identifying common factors with the denominator. If there are any common factors, we can cancel them out, which simplifies the expression. Factoring the numerator is a key step in simplifying rational expressions because it allows us to reduce the expression to its simplest form. A factored numerator makes it easier to identify and cancel common factors with the denominator. This is a fundamental step in simplifying rational expressions and is a prerequisite for the final simplification. Factoring the numerator is not always straightforward. It may require the application of multiple factoring techniques and a good understanding of algebraic patterns. However, the effort is worthwhile because it can lead to significant simplifications. The factored form of the numerator provides valuable information about the expression. It reveals the roots of the numerator, which can be useful in various contexts, such as solving equations or analyzing the behavior of the expression. However, in the context of simplifying rational expressions, the primary benefit of factoring is to identify common factors that can be canceled out. In this specific case, factoring out -5 simplifies the expression and makes it easier to factor the remaining quadratic expression. This step demonstrates the importance of looking for common factors first, as it can often simplify the factoring process. Factoring is not just a mechanical process; it's a way of gaining deeper insight into the structure of an expression. By factoring, we are essentially breaking down the expression into its fundamental components, which can reveal hidden relationships and simplifications.

Step 7: Simplifying the Rational Expression

Now we have the expression (-5(x^2 - 6x - 21)) / (6(x - 3)^2). At this point, we need to check if there are any common factors between the numerator and the denominator that can be canceled out. In this case, there are no common factors. The quadratic expression (x^2 - 6x - 21) does not factor easily and does not share any factors with (x - 3)^2. Therefore, the simplified rational expression is (-5(x^2 - 6x - 21)) / (6(x - 3)^2). Simplifying rational expressions involves identifying and canceling common factors between the numerator and the denominator. This is the final step in reducing the expression to its simplest form. It requires a careful analysis of the factored numerator and denominator to identify any factors that appear in both. In the context of rational expressions, canceling common factors is a crucial step because it eliminates redundant terms and simplifies the expression. This makes the expression easier to work with and understand. The process of simplifying rational expressions involves careful attention to the factors in the numerator and the denominator. We need to identify factors that are exactly the same and then cancel them out. This may require multiple steps and careful attention to detail. The simplified rational expression represents the original expression in its most reduced form. This form is crucial for various applications, such as solving equations, graphing functions, and analyzing the behavior of the expression. Simplifying rational expressions is a key step in algebraic manipulations because it makes the expression easier to work with. A simplified expression is less prone to errors and is easier to understand. This is particularly important in more complex problems where the expression may be just one part of a larger calculation. The absence of common factors indicates that the expression is already in its simplest form. This means that we cannot reduce the expression further by canceling out any factors. The simplified expression may still contain complex terms, but it is in its most reduced form. In this specific case, the expression (-5(x^2 - 6x - 21)) / (6(x - 3)^2) is the simplified form of the original expression. This means that we have successfully combined the two fractions into a single, simplified rational expression. This step demonstrates the importance of checking for common factors after factoring the numerator and the denominator. It ensures that we have reduced the expression to its simplest form and that there are no redundant terms. Simplifying rational expressions is a fundamental skill in algebra and is a prerequisite for many other algebraic manipulations. It requires a good understanding of factoring, common factors, and the rules of fraction arithmetic. This skill is essential for solving equations, graphing functions, and working with complex algebraic expressions. This final step highlights the importance of being thorough and meticulous in algebraic manipulations. By carefully checking for common factors, we ensure that we have arrived at the simplest possible form of the expression.

Conclusion

Simplifying rational expressions is a multi-step process that requires a strong foundation in algebraic techniques. By following these steps – factoring the denominator, finding the LCD, rewriting fractions, combining numerators, expanding and simplifying, factoring the numerator, and simplifying the expression – you can confidently tackle these types of problems. Mastering these skills is essential for success in algebra and beyond. The process of simplifying rational expressions is not just about applying a set of rules; it's about developing a deeper understanding of algebraic structures and relationships. Each step in the process builds upon the previous one, and a clear understanding of each step is crucial for success. This article has provided a detailed guide to simplifying a specific rational expression, but the principles and techniques discussed can be applied to a wide range of similar problems. The key is to approach each problem systematically and to pay close attention to the details. Algebra is a building-block subject, and mastering the fundamentals is essential for tackling more advanced topics. Simplifying rational expressions is one of those fundamental skills that will serve you well in future math courses and in various applications of mathematics. This article has aimed to provide a clear and comprehensive guide to this skill, but practice is essential for mastery. Work through additional examples, and don't be afraid to ask for help when you need it. The effort you put in will pay off in a deeper understanding of algebra and a greater confidence in your problem-solving abilities. In conclusion, simplifying rational expressions is a valuable skill that requires a combination of algebraic techniques and careful attention to detail. By following the steps outlined in this article and practicing regularly, you can master this skill and enhance your overall algebraic proficiency. This will not only help you in your math courses but also in various other fields where algebraic thinking is essential. Remember, the key to success in algebra is to build a strong foundation in the fundamentals and to practice consistently. With dedication and effort, you can master the art of simplifying rational expressions and unlock new levels of algebraic understanding.