Factoring Quadratics A Step-by-Step Guide To Factoring 8x² - 50
Factoring quadratic expressions is a fundamental skill in algebra, allowing us to simplify expressions, solve equations, and gain deeper insights into mathematical relationships. When faced with the expression 8x² - 50, the task of finding its completely factored form might seem daunting at first. However, by systematically applying factoring techniques, we can unravel the structure of this expression and arrive at its simplest representation. This article serves as a comprehensive guide, meticulously walking you through the steps involved in factoring 8x² - 50 completely. We will explore the underlying principles of factoring, identify common factors, apply the difference of squares pattern, and ultimately arrive at the correct factored form. Whether you're a student grappling with algebra concepts or simply seeking to enhance your mathematical prowess, this guide will empower you with the knowledge and skills to confidently tackle factoring problems.
Understanding the Basics of Factoring
Before diving into the specifics of factoring 8x² - 50, let's lay a solid foundation by revisiting the fundamental concepts of factoring. At its core, factoring is the process of breaking down a mathematical expression into a product of its constituent parts, known as factors. These factors, when multiplied together, yield the original expression. Think of it as the reverse operation of expanding or distributing. For instance, consider the expression 12. We can factor it as 2 × 6, 3 × 4, or even 2 × 2 × 3. Each of these representations expresses 12 as a product of its factors. In algebra, we extend this concept to expressions involving variables. For example, the expression x² + 5x + 6 can be factored as (x + 2)(x + 3). When we expand the factored form, we get back the original expression. Factoring is a powerful tool because it allows us to rewrite expressions in a more manageable form, making it easier to solve equations, simplify fractions, and analyze mathematical relationships. Different factoring techniques exist, each tailored to specific types of expressions. We'll explore some of these techniques as we delve into factoring 8x² - 50. The key principle to remember is that factoring aims to express an expression as a product of its factors, unveiling its underlying structure. By mastering factoring, you unlock a deeper understanding of algebraic expressions and their behavior.
Identifying the Greatest Common Factor (GCF)
The first and often most crucial step in factoring any expression is to identify and extract the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all terms of the expression. In the case of 8x² - 50, we need to find the largest factor that divides both 8x² and 50. Let's break down each term individually. The factors of 8x² are 1, 2, 4, 8, x, x², 2x, 4x, 8x, 2x², 4x², and 8x². The factors of 50 are 1, 2, 5, 10, 25, and 50. Comparing the factors of both terms, we can see that the largest factor they share is 2. Therefore, the GCF of 8x² and 50 is 2. Now, we can factor out the GCF from the expression. Factoring out 2 from 8x² - 50 gives us 2(4x² - 25). This means we divide each term in the original expression by 2 and write the result inside the parentheses. Factoring out the GCF is a crucial step because it simplifies the expression and often reveals further factoring opportunities. In this case, after factoring out the GCF, we are left with 4x² - 25, which has a recognizable form that we can factor further. Always remember to look for the GCF as the first step in any factoring problem. It can significantly simplify the process and lead you to the completely factored form.
Recognizing the Difference of Squares Pattern
After extracting the GCF from 8x² - 50, we arrived at the expression 2(4x² - 25). Now, we need to examine the expression inside the parentheses, 4x² - 25, to see if it can be factored further. This is where recognizing patterns becomes essential. In this case, we can observe that 4x² and 25 are both perfect squares. 4x² is the square of 2x, and 25 is the square of 5. Furthermore, the two terms are separated by a subtraction sign. This pattern, where we have the difference of two perfect squares, is a classic factoring scenario known as the difference of squares. The difference of squares pattern states that a² - b² can be factored as (a + b)(a - b). This pattern provides a shortcut for factoring expressions of this form. To apply the difference of squares pattern to 4x² - 25, we need to identify what 'a' and 'b' represent in our expression. As we noted earlier, 4x² is the square of 2x, so 'a' is 2x. Similarly, 25 is the square of 5, so 'b' is 5. Now, we can directly apply the difference of squares pattern: 4x² - 25 = (2x + 5)(2x - 5). This factorization expresses 4x² - 25 as the product of two binomials, (2x + 5) and (2x - 5). Recognizing the difference of squares pattern is a valuable skill in factoring. It allows you to quickly factor expressions that fit this pattern, saving you time and effort. In the next step, we will combine this factorization with the GCF we extracted earlier to arrive at the completely factored form of 8x² - 50.
Completing the Factorization
We've made significant progress in factoring 8x² - 50. First, we identified and extracted the greatest common factor (GCF), which was 2, giving us 2(4x² - 25). Then, we recognized the difference of squares pattern in the expression 4x² - 25 and factored it as (2x + 5)(2x - 5). Now, to obtain the completely factored form of the original expression, we need to combine these two factorizations. We simply bring the GCF, 2, back into the picture and multiply it by the factored form of 4x² - 25. This gives us: 2(2x + 5)(2x - 5). This is the completely factored form of 8x² - 50. It expresses the original expression as a product of three factors: 2, (2x + 5), and (2x - 5). To verify that our factorization is correct, we can expand the factored form and see if it matches the original expression. Expanding 2(2x + 5)(2x - 5), we first multiply (2x + 5) and (2x - 5) using the distributive property or the FOIL method: (2x + 5)(2x - 5) = 4x² - 10x + 10x - 25 = 4x² - 25. Then, we multiply the result by 2: 2(4x² - 25) = 8x² - 50. This confirms that our factorization is indeed correct. The completely factored form, 2(2x + 5)(2x - 5), represents the simplest way to express 8x² - 50 as a product of its factors. It reveals the underlying structure of the expression and can be useful for solving equations, simplifying expressions, and analyzing mathematical relationships.
Selecting the Correct Answer
Now that we have successfully factored 8x² - 50 completely, let's revisit the answer choices provided and identify the correct one. We found that the completely factored form is 2(2x + 5)(2x - 5). Comparing this with the given options:
A. 2(x + 5)(x - 5) B. 2(2x - 5)(2x - 5) C. 2(2x + 5)(2x + 5) D. 2(2x + 5)(2x - 5)
We can see that option D, 2(2x + 5)(2x - 5), matches our result exactly. Therefore, option D is the correct answer. The other options are incorrect because they do not represent the completely factored form of 8x² - 50. Option A is missing the coefficients of x within the parentheses. Options B and C have incorrect signs within the parentheses, leading to a different expanded form. Selecting the correct answer is the final step in the factoring process. It demonstrates that you have successfully applied the factoring techniques and arrived at the correct representation of the expression. Always double-check your work and compare your result with the given options to ensure accuracy. By carefully following the steps outlined in this guide, you can confidently factor quadratic expressions and select the correct answer.
Conclusion: Mastering Factoring Techniques
In this comprehensive guide, we have successfully navigated the process of factoring the quadratic expression 8x² - 50 completely. We began by establishing a solid understanding of the basics of factoring, emphasizing its role in simplifying expressions and revealing their underlying structure. We then systematically applied factoring techniques, starting with identifying and extracting the greatest common factor (GCF), which was 2. This crucial step simplified the expression and paved the way for further factorization. Next, we recognized the difference of squares pattern in the expression 4x² - 25, allowing us to factor it as (2x + 5)(2x - 5). Finally, we combined the GCF with the factored form obtained from the difference of squares pattern, arriving at the completely factored form: 2(2x + 5)(2x - 5). Throughout this process, we emphasized the importance of recognizing patterns and applying appropriate factoring techniques. We also highlighted the significance of verifying the factorization by expanding the factored form and comparing it with the original expression. Mastering factoring techniques is essential for success in algebra and beyond. It empowers you to simplify expressions, solve equations, and gain deeper insights into mathematical relationships. By practicing these techniques and applying them consistently, you can develop confidence and proficiency in factoring quadratic expressions and other algebraic forms. Remember, factoring is not just a mechanical process; it's a way of understanding the structure and behavior of mathematical expressions. With a solid foundation in factoring, you'll be well-equipped to tackle more advanced mathematical concepts and challenges.
