Identifying Common Factors Of Quadratic Expressions An In-Depth Solution For X²-9 And X²+8x+15

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In this article, we will delve into the problem of finding a common factor between two quadratic expressions: x² - 9 and x² + 8x + 15. This is a fundamental concept in algebra, often encountered in high school mathematics. Understanding how to factor quadratic expressions and identify common factors is crucial for simplifying algebraic expressions, solving equations, and grasping more advanced mathematical concepts. We will explore the step-by-step process of factoring each expression and then pinpoint the factor that appears in both. This exploration will not only solve the given problem but also reinforce the underlying principles of factoring and algebraic manipulation. The ability to identify and extract common factors is a cornerstone of algebraic proficiency, enabling students and practitioners to tackle a wide array of mathematical challenges with confidence and precision.

Factoring x² - 9

To begin, let's focus on factoring the first expression: x² - 9. This expression is a classic example of a difference of squares. The difference of squares pattern is a fundamental algebraic identity that states a² - b² can be factored into (a + b)(a - b). Recognizing this pattern is the first step in efficiently factoring such expressions. In our case, x² can be seen as the square of x, and 9 can be seen as the square of 3 (since 3² = 9). Applying the difference of squares pattern, we can rewrite x² - 9 as x² - 3². Now, we can directly apply the formula a² - b² = (a + b)(a - b), where a is x and b is 3. This gives us the factored form (x + 3)(x - 3). Therefore, the expression x² - 9 is factored into two binomials: (x + 3) and (x - 3). These are the two factors of the original expression. Understanding and applying the difference of squares pattern is a valuable skill in algebra, as it allows for quick and efficient factorization of expressions in this form. This ability is not only crucial for simplifying expressions but also for solving equations and tackling more complex algebraic problems.

Factoring x² + 8x + 15

Next, we turn our attention to the second expression: x² + 8x + 15. This is a quadratic trinomial, and we need to factor it into two binomials. Factoring a quadratic trinomial of the form ax² + bx + c involves finding two numbers that multiply to c (the constant term) and add up to b (the coefficient of the x term). In our case, we are looking for two numbers that multiply to 15 and add up to 8. Let's list the factors of 15: 1 and 15, 3 and 5. Among these pairs, 3 and 5 satisfy both conditions: 3 * 5 = 15 and 3 + 5 = 8. Therefore, we can use these numbers to factor the trinomial. We can rewrite the middle term, 8x, as 3x + 5x. This gives us x² + 3x + 5x + 15. Now we can factor by grouping. Group the first two terms and the last two terms: (x² + 3x) + (5x + 15). Factor out the greatest common factor (GCF) from each group. From the first group, we can factor out x, resulting in x(x + 3). From the second group, we can factor out 5, resulting in 5(x + 3). Now we have x(x + 3) + 5(x + 3). Notice that (x + 3) is a common factor in both terms. We can factor out (x + 3), which leaves us with (x + 3)(x + 5). Thus, the expression x² + 8x + 15 factors into two binomials: (x + 3) and (x + 5). This method of factoring quadratic trinomials is a fundamental skill in algebra, essential for simplifying expressions and solving quadratic equations.

Identifying the Common Factor

Now that we have factored both expressions, we can identify the common factor. We factored x² - 9 into (x + 3)(x - 3) and x² + 8x + 15 into (x + 3)(x + 5). Comparing the two factored forms, we can see that the binomial (x + 3) appears in both. This means that (x + 3) is a common factor of both expressions. Common factors are crucial in simplifying algebraic expressions and solving equations. Identifying them allows us to reduce complex expressions to simpler forms, making them easier to work with. In this case, finding the common factor (x + 3) helps us understand the relationship between the two quadratic expressions. It highlights a shared component that is essential for further algebraic manipulations, such as solving simultaneous equations or simplifying rational expressions. Recognizing and extracting common factors is a fundamental skill in algebra, providing a powerful tool for problem-solving and algebraic manipulation. This skill is not only applicable to quadratic expressions but extends to various types of algebraic problems, making it a cornerstone of algebraic proficiency.

Conclusion

In conclusion, by factoring both expressions x² - 9 and x² + 8x + 15, we were able to identify the common factor. The expression x² - 9 factors into (x + 3)(x - 3) using the difference of squares pattern. The expression x² + 8x + 15 factors into (x + 3)(x + 5) using the factoring by grouping method. Comparing the factored forms, the common factor is (x + 3). Therefore, the correct answer is B. This exercise underscores the importance of mastering factoring techniques in algebra. Being able to factor expressions efficiently allows for the simplification of complex algebraic problems and provides a foundation for more advanced mathematical concepts. The ability to recognize patterns, such as the difference of squares, and to apply factoring methods, such as factoring by grouping, are essential skills for success in algebra and beyond. Furthermore, the process of identifying common factors is a crucial step in simplifying expressions and solving equations, making it a fundamental aspect of algebraic proficiency. By practicing these skills, students can develop a deeper understanding of algebraic principles and improve their problem-solving abilities.