Factoring 6x² - 13x - 5 A Step-by-Step Solution
Factoring quadratic expressions is a fundamental skill in algebra, and understanding the process allows us to solve equations, simplify expressions, and gain deeper insights into mathematical relationships. In this comprehensive guide, we will delve into the world of factoring, specifically focusing on the quadratic expression 6x² - 13x - 5. We'll break down the steps involved in finding its completely factored form, explore the underlying concepts, and equip you with the knowledge to tackle similar problems with confidence. We will navigate through the intricacies of factoring, providing clear explanations and illustrative examples along the way. By the end of this guide, you'll not only know the answer to this specific problem but also possess a strong foundation in factoring techniques that you can apply to a wide range of algebraic challenges. Factoring is like unlocking a secret code, revealing the hidden structure within an expression. It's a powerful tool that opens doors to more advanced mathematical concepts, making it an essential skill for anyone pursuing further studies in mathematics, science, or engineering. This guide is designed to be accessible to learners of all levels, whether you're just starting your algebraic journey or looking to sharpen your factoring skills. So, let's embark on this factoring adventure together and unravel the completely factored form of 6x² - 13x - 5.
Understanding Factoring and Quadratic Expressions
Before diving into the specific problem, it's crucial to understand the core concepts of factoring and quadratic expressions. Factoring, in essence, is the process of breaking down an expression into its constituent parts, typically represented as a product of simpler expressions. Think of it like reversing the distributive property – instead of multiplying factors together, we're identifying the factors that, when multiplied, produce the original expression. In the realm of factoring, quadratic expressions hold a special significance. A quadratic expression is a polynomial of degree two, generally expressed in the standard form of ax² + bx + c, where a, b, and c are constants, and 'a' is not equal to zero. The 'x²' term is the hallmark of a quadratic expression, setting it apart from linear expressions (degree one) or cubic expressions (degree three). The coefficients 'a', 'b', and 'c' play crucial roles in determining the behavior and properties of the quadratic expression. The coefficient 'a' dictates the parabola's direction (upward if positive, downward if negative) and its width. The coefficient 'b' influences the parabola's horizontal position, and the constant 'c' represents the y-intercept, the point where the parabola intersects the vertical axis. Factoring quadratic expressions allows us to rewrite them in a more manageable form, often as a product of two binomials (expressions with two terms). This transformation is incredibly useful for solving quadratic equations, finding the roots (x-intercepts) of the quadratic function, and simplifying complex algebraic expressions. The ability to factor quadratic expressions effectively is a cornerstone of algebraic manipulation, and it lays the foundation for more advanced mathematical concepts, such as solving systems of equations, working with rational expressions, and understanding the behavior of polynomial functions. We often use factoring in everyday situations, from calculating areas to understanding financial growth. By mastering the art of factoring, you'll not only enhance your mathematical skills but also gain a valuable tool for problem-solving in various contexts.
The Target Expression: 6x² - 13x - 5
Let's turn our attention to the specific quadratic expression we aim to factor: 6x² - 13x - 5. This expression adheres to the standard quadratic form, ax² + bx + c, where a = 6, b = -13, and c = -5. These coefficients will guide us through the factoring process. The coefficient 'a' (6) indicates that the leading term is 6x², which means our factored form will likely involve terms with 'x' that, when multiplied, result in 6x². The negative coefficient 'b' (-13) suggests that the middle term (-13x) arises from a combination of products involving 'x' terms and constants within the binomial factors. The negative constant term 'c' (-5) implies that the constant terms in our binomial factors will have opposite signs, since their product is negative. The challenge lies in finding the correct combination of factors that, when expanded, yield the original expression, 6x² - 13x - 5. This process often involves a bit of trial and error, guided by our understanding of the coefficients and their relationships. We'll explore different techniques for factoring this expression, each with its own strengths and approach. The goal is not just to find the correct factored form but also to understand the reasoning behind each step, empowering you to factor similar expressions with confidence. By dissecting this expression, we'll uncover the hidden factors that compose it, revealing the underlying structure that governs its behavior. This process is akin to detective work, where we use clues and logical deduction to arrive at the solution. So, let's put on our mathematical detective hats and begin the quest to factor 6x² - 13x - 5!
Method 1: The AC Method
The AC method is a powerful technique for factoring quadratic expressions, especially when the leading coefficient (a) is not equal to 1. It provides a systematic approach to breaking down the expression and identifying the correct factors. The core idea behind the AC method is to focus on the product of the coefficients 'a' and 'c', hence the name "AC method." In our case, with the expression 6x² - 13x - 5, a = 6 and c = -5, so AC = 6 * (-5) = -30. The next crucial step is to find two numbers that multiply to AC (-30) and add up to the coefficient 'b' (-13). This is where the detective work comes in. We need to consider the factors of -30 and see which pair sums to -13. After some thought, we can identify the pair as -15 and 2. Indeed, -15 * 2 = -30 and -15 + 2 = -13. With these two numbers in hand, we rewrite the middle term (-13x) as the sum of two terms using these numbers as coefficients: -15x + 2x. This transforms our expression from 6x² - 13x - 5 to 6x² - 15x + 2x - 5. Now, we have four terms, and we can proceed to factor by grouping. We group the first two terms and the last two terms: (6x² - 15x) + (2x - 5). Next, we factor out the greatest common factor (GCF) from each group. From the first group, 6x² - 15x, the GCF is 3x, leaving us with 3x(2x - 5). From the second group, 2x - 5, the GCF is 1, so we have 1(2x - 5). Notice that both groups now share a common binomial factor: (2x - 5). We can factor out this common binomial factor, resulting in (2x - 5)(3x + 1). And there you have it! We've successfully factored 6x² - 13x - 5 using the AC method.
Method 2: Trial and Error (Intelligent Guessing)
While the AC method provides a structured approach, another effective technique for factoring quadratic expressions is trial and error, often referred to as intelligent guessing. This method relies on our understanding of how binomial factors multiply to produce a quadratic expression and involves strategically testing different combinations until we find the correct one. To apply the trial and error method to 6x² - 13x - 5, we start by considering the possible factors of the leading term (6x²) and the constant term (-5). The factors of 6x² are 6x and x, or 3x and 2x. The factors of -5 are 5 and -1, or -5 and 1. Our goal is to arrange these factors into two binomials such that their product equals the original quadratic expression. We can start by trying the combination (2x )(3x ). We know that the constant terms in the binomials must multiply to -5, so we have two possibilities to consider: (2x + 5)(3x - 1) or (2x - 5)(3x + 1). Let's test the first possibility, (2x + 5)(3x - 1), by expanding it using the distributive property (or the FOIL method): (2x + 5)(3x - 1) = 6x² - 2x + 15x - 5 = 6x² + 13x - 5. This doesn't match our target expression, 6x² - 13x - 5, because the middle term has the wrong sign. This tells us we're on the right track but need to adjust the signs. Let's try the second possibility, (2x - 5)(3x + 1): (2x - 5)(3x + 1) = 6x² + 2x - 15x - 5 = 6x² - 13x - 5. This matches our original expression perfectly! Therefore, the completely factored form of 6x² - 13x - 5 is (2x - 5)(3x + 1). The key to successful trial and error is to be systematic and keep track of the combinations you've tried. It also helps to have a good understanding of the relationships between the coefficients and the factors.
The Completely Factored Form: (2x - 5)(3x + 1)
After employing both the AC method and the trial and error technique, we've arrived at the same conclusion: the completely factored form of 6x² - 13x - 5 is (2x - 5)(3x + 1). This result represents the expression 6x² - 13x - 5 broken down into its fundamental multiplicative components. Each binomial factor, (2x - 5) and (3x + 1), is a linear expression, and their product yields the original quadratic expression. This factorization is complete because neither of the binomial factors can be factored further. They are in their simplest form. To verify our result, we can expand the factored form using the distributive property (or the FOIL method): (2x - 5)(3x + 1) = 2x * 3x + 2x * 1 - 5 * 3x - 5 * 1 = 6x² + 2x - 15x - 5 = 6x² - 13x - 5. The expansion confirms that our factored form is indeed equivalent to the original expression, 6x² - 13x - 5. The completely factored form provides valuable insights into the behavior of the quadratic expression. For instance, it allows us to easily find the roots (or zeros) of the corresponding quadratic equation, 6x² - 13x - 5 = 0. The roots are the values of 'x' that make the equation true, and they correspond to the x-intercepts of the parabola represented by the quadratic expression. Setting each factor to zero, we get: 2x - 5 = 0, which gives x = 5/2, and 3x + 1 = 0, which gives x = -1/3. These are the roots of the quadratic equation and the x-intercepts of its graph. In conclusion, the completely factored form (2x - 5)(3x + 1) is not just a mathematical solution; it's a key that unlocks a deeper understanding of the quadratic expression and its properties.
Answer and Options Analysis
Now that we've successfully factored the quadratic expression 6x² - 13x - 5 and determined the completely factored form to be (2x - 5)(3x + 1), let's analyze the provided options and identify the correct answer. We were given the following options:
A. (2x - 5)(3x + 1) B. (2x + 5)(3x - 1) C. (2x - 1)(3x - 5) D. (2x + 1)(3x + 5)
Comparing our result, (2x - 5)(3x + 1), with the options, we can clearly see that option A, (2x - 5)(3x + 1), matches our solution perfectly. Therefore, option A is the correct answer. To further solidify our understanding, let's examine why the other options are incorrect. Option B, (2x + 5)(3x - 1), when expanded, yields 6x² + 13x - 5, which has the correct leading and constant terms but the wrong sign for the middle term. Option C, (2x - 1)(3x - 5), expands to 6x² - 13x + 5, which has the correct leading and middle terms but the wrong sign for the constant term. Option D, (2x + 1)(3x + 5), expands to 6x² + 13x + 5, which has the correct leading term but the wrong signs for both the middle and constant terms. This analysis highlights the importance of carefully checking the signs and coefficients when factoring quadratic expressions. A small error in sign can lead to an incorrect factorization. By systematically working through the problem and verifying our result, we can confidently identify the correct answer and understand why the other options are not viable. In summary, the correct answer is A. (2x - 5)(3x + 1), as it is the only option that, when expanded, yields the original quadratic expression, 6x² - 13x - 5.
Conclusion: Mastering Factoring for Algebraic Success
In this comprehensive guide, we've successfully navigated the process of factoring the quadratic expression 6x² - 13x - 5, revealing its completely factored form: (2x - 5)(3x + 1). We explored two powerful techniques – the AC method and trial and error – each offering a unique approach to tackling factoring problems. The AC method provided a structured, step-by-step process, while trial and error allowed us to leverage our understanding of binomial multiplication to arrive at the solution. By applying these methods and carefully analyzing the coefficients and signs, we confidently determined the correct factorization. Factoring quadratic expressions is a cornerstone of algebra, and mastering this skill opens doors to a wide range of mathematical applications. From solving quadratic equations to simplifying complex expressions, factoring empowers us to manipulate and understand algebraic relationships more effectively. The ability to factor is not just about finding the right answer; it's about developing a deeper understanding of the structure and behavior of mathematical expressions. It's about seeing the connections between different concepts and building a solid foundation for future learning. As you continue your mathematical journey, remember that practice is key. The more you factor, the more comfortable and confident you'll become. Experiment with different techniques, challenge yourself with increasingly complex expressions, and don't be afraid to make mistakes – they're valuable learning opportunities. By embracing the challenges and celebrating the successes, you'll unlock the power of factoring and achieve greater algebraic success. So, keep factoring, keep exploring, and keep building your mathematical prowess!
