Factors Of M^2 - 14m + 48: A Step-by-Step Solution
Unraveling quadratic expressions is a fundamental skill in algebra, and understanding how to factor them is crucial for solving equations, simplifying expressions, and tackling more advanced mathematical concepts. In this comprehensive guide, we will dissect the quadratic expression m^2 - 14m + 48 and meticulously explore the process of identifying its factors. Our focus will be on a step-by-step approach, ensuring clarity and understanding for learners of all levels. The goal is not just to find the correct answer but to grasp the underlying principles of factorization. Before diving into the specifics of this particular expression, let's briefly review the basics of factoring quadratic expressions. A quadratic expression is a polynomial of degree two, generally written in the form ax^2 + bx + c, where a, b, and c are constants. Factoring a quadratic expression means rewriting it as a product of two linear expressions. This is the reverse process of expanding brackets, and it relies on identifying two numbers that satisfy certain conditions related to the coefficients of the quadratic expression. In our case, the expression is m^2 - 14m + 48, where a = 1, b = -14, and c = 48. The key to factoring this lies in finding two numbers that multiply to 48 (the constant term) and add up to -14 (the coefficient of the m term). This is a classic factorization problem, and once we identify these two numbers, we can easily write down the factors of the quadratic expression. Let's embark on this journey of factorization, and by the end of this guide, you'll not only know the factors of m^2 - 14m + 48 but also the 'why' behind the process.
The Quest for the Right Numbers
In this section, we delve deep into the heart of factoring the quadratic expression m^2 - 14m + 48, focusing on the pivotal step of finding the two numbers that dictate the factors. As we established, the magic lies in identifying two numbers that multiply to the constant term (48) and add up to the coefficient of the linear term (-14). This might seem like a daunting task initially, but with a systematic approach and a touch of mathematical intuition, it becomes a manageable and even enjoyable puzzle. We begin by systematically listing the pairs of factors of 48. It's crucial to consider both positive and negative factors since the product is positive (48), but the sum is negative (-14). This immediately tells us that both numbers must be negative. The factor pairs of 48 are (1, 48), (2, 24), (3, 16), (4, 12), and (6, 8). Now, we need to consider their negative counterparts: (-1, -48), (-2, -24), (-3, -16), (-4, -12), and (-6, -8). The next step is to examine the sums of these negative factor pairs. We are looking for a pair that adds up to -14. Adding the pairs, we get: -1 + (-48) = -49, -2 + (-24) = -26, -3 + (-16) = -19, -4 + (-12) = -16, and -6 + (-8) = -14. Bingo! The pair -6 and -8 perfectly fits our criteria. They multiply to 48 and add up to -14. This is a critical juncture in the factoring process. Having identified these two numbers, we are now on the cusp of expressing the quadratic expression as a product of two binomials. Understanding this step is not just about solving this particular problem; it's about building a solid foundation for factoring any quadratic expression. With -6 and -8 in hand, the next step is to construct the factors, which will lead us to the correct answer among the given options.
Constructing the Factors: The Solution Unveiled
Having successfully identified -6 and -8 as the two crucial numbers, we are now poised to construct the factors of the quadratic expression m^2 - 14m + 48. This step is where the abstract concept of factoring transforms into a concrete expression, revealing the underlying structure of the quadratic. The two numbers we found, -6 and -8, directly translate into the constant terms within the binomial factors. Since our variable is 'm', the factors will take the form (m + a)(m + b), where 'a' and 'b' are the numbers we found. In this case, a = -6 and b = -8. Therefore, the factors of m^2 - 14m + 48 can be written as (m - 6)(m - 8). This is a significant moment in the solution process. We have transformed the quadratic expression into its factored form, which is a product of two linear expressions. It's important to understand the connection between the numbers we found and the factors we constructed. The numbers -6 and -8 are the roots of the quadratic equation m^2 - 14m + 48 = 0. This means that if we were to solve this equation, the solutions would be m = 6 and m = 8. This relationship between factors and roots is a fundamental concept in algebra and is crucial for understanding the behavior of quadratic functions. Now that we have the factors, we can compare them to the given options to identify the correct answer. The options provided are: A) (m - 12)(m + 4), B) (m - 12)(m - 4), C) (m - 6)(m - 8), and D) (m + 6)(m + 8). By direct comparison, it's clear that option C, (m - 6)(m - 8), matches the factors we derived. This confirms that we have successfully factored the quadratic expression. But our journey doesn't end here. It's crucial to verify our solution to ensure accuracy and deepen our understanding of the factoring process.
Verifying the Solution: Ensuring Accuracy and Understanding
In the realm of mathematics, verification is as crucial as the solution itself. It's the process that solidifies our understanding and ensures the accuracy of our results. For the quadratic expression m^2 - 14m + 48, we've identified (m - 6)(m - 8) as the factors. To verify this, we'll perform the reverse operation of factoring, which is expansion. Expanding the product of the two binomials (m - 6)(m - 8) should yield the original quadratic expression if our factoring is correct. The expansion process involves multiplying each term in the first binomial by each term in the second binomial. This is often remembered by the acronym FOIL, which stands for First, Outer, Inner, Last. Applying FOIL to (m - 6)(m - 8), we get:
- First: m * m = m^2
- Outer: m * -8 = -8m
- Inner: -6 * m = -6m
- Last: -6 * -8 = 48
Adding these terms together, we have m^2 - 8m - 6m + 48. Combining the like terms (-8m and -6m), we get m^2 - 14m + 48. This is precisely the original quadratic expression we started with! This verification step is not just a formality; it's a powerful way to confirm our understanding of the factoring process. It demonstrates that we can move seamlessly between the factored form and the expanded form of a quadratic expression. Moreover, verification helps catch any potential errors in our calculations or reasoning. If the expanded form did not match the original expression, it would signal a mistake in our factoring process, prompting us to revisit our steps. In this case, our verification confirms that (m - 6)(m - 8) are indeed the correct factors of m^2 - 14m + 48. With this verification complete, we can confidently assert that option C is the correct answer. But beyond the specific answer, the process of factoring and verifying has provided us with a deeper understanding of quadratic expressions and their properties.
Conclusion: Mastering the Art of Factoring
Our journey through factoring the quadratic expression m^2 - 14m + 48 has been more than just finding the correct answer; it has been an exploration of the fundamental principles of algebra. We began by understanding the basics of factoring quadratic expressions, emphasizing the importance of finding two numbers that satisfy specific conditions related to the coefficients. We then embarked on the quest for these numbers, systematically identifying the factor pairs of the constant term and their sums, ultimately pinpointing -6 and -8 as the key. These numbers paved the way for constructing the factors (m - 6)(m - 8), which directly led us to the correct answer, option C. However, our exploration didn't stop there. We recognized the crucial role of verification in ensuring accuracy and solidifying understanding. By expanding the factors, we confirmed that they indeed produced the original quadratic expression, reinforcing our grasp of the factoring process. The ability to factor quadratic expressions is a cornerstone of algebraic manipulation. It's a skill that transcends this specific problem and applies to a wide range of mathematical contexts. From solving quadratic equations to simplifying rational expressions, factoring is an indispensable tool. Mastering this art requires not just memorizing steps but understanding the underlying logic and connections. The relationship between factors and roots, the process of expansion and its reverse, the systematic search for the right numbers – these are all pieces of the puzzle that come together to form a comprehensive understanding of factoring. As you continue your mathematical journey, remember that practice is the key to mastery. The more you engage with factoring problems, the more intuitive the process will become. So, embrace the challenge, explore different quadratic expressions, and watch your factoring skills flourish. And always remember the power of verification – it's the ultimate safeguard against errors and the gateway to deeper understanding.
