Factoring Quadratics Finding Factors Of M^2 - 14m + 48
Introduction: Understanding Quadratic Expressions and Factoring
In the realm of algebra, quadratic expressions play a pivotal role. These expressions, characterized by a variable raised to the power of two, frequently appear in various mathematical and real-world problems. Factoring, a fundamental algebraic technique, involves breaking down a quadratic expression into its constituent factors. This process not only simplifies the expression but also provides valuable insights into its roots and behavior. In this comprehensive guide, we will delve into the art of factoring quadratic expressions, specifically focusing on the expression m^2 - 14m + 48. We will dissect the expression, explore different factoring methods, and ultimately identify the correct factors from the given options. Mastering factoring techniques is crucial for success in algebra and beyond, as it lays the foundation for solving equations, graphing functions, and tackling more advanced mathematical concepts.
Factoring is the reverse process of expanding. When we expand an expression like (x + a)(x + b), we use the distributive property to multiply each term in the first parenthesis by each term in the second parenthesis, resulting in a quadratic expression. Factoring, on the other hand, starts with the quadratic expression and aims to find the original binomial factors. This process is akin to reverse engineering, where we try to reconstruct the original components from the final product. Understanding this relationship between expansion and factoring is essential for mastering the technique. The ability to factor quadratic expressions efficiently is not just a mathematical skill; it's a problem-solving tool that can be applied in various fields, from physics and engineering to economics and computer science. By mastering factoring, you equip yourself with a powerful technique to tackle a wide range of problems.
Before we dive into the specifics of factoring m^2 - 14m + 48, let's briefly touch upon the general form of a quadratic expression and the terminology associated with it. A quadratic expression is typically written in the form ax^2 + bx + c, where a, b, and c are constants, and x is the variable. The term ax^2 is the quadratic term, bx is the linear term, and c is the constant term. In the expression m^2 - 14m + 48, a is 1, b is -14, and c is 48. Understanding these coefficients is crucial for factoring the expression correctly. Factoring involves finding two binomials that, when multiplied together, produce the original quadratic expression. These binomials are the factors of the quadratic expression. The goal is to find the right combination of numbers that satisfy the conditions imposed by the coefficients of the quadratic expression. This requires a systematic approach and a keen eye for patterns.
Problem Statement: Identifying Factors of m^2 - 14m + 48
Our primary objective is to determine the correct factors of the quadratic expression m^2 - 14m + 48. We are presented with four options, each consisting of a pair of binomials. To solve this problem effectively, we will employ the principles of factoring quadratic expressions and systematically evaluate each option. The options are:
- A) $(m-12)(m-4)$
- B) $(m-12)(m+4)$
- C) $(m-6)(m-8)$
- D) $(m+6)(m+8)$
To solve this problem, we must identify the factors that, when multiplied, yield the original quadratic expression. This involves understanding the relationship between the coefficients of the quadratic expression and the constants within the binomial factors. The constant term of the quadratic expression (48 in this case) is the product of the constant terms in the binomial factors. The coefficient of the linear term (-14m in this case) is the sum of the products of the constant terms in the binomial factors with the variable m. By carefully analyzing these relationships, we can narrow down the possibilities and identify the correct factors. This process requires a methodical approach and a clear understanding of the factoring principles.
Factoring quadratic expressions is a skill that requires practice and attention to detail. There are several methods for factoring, including trial and error, using the quadratic formula, and completing the square. However, for this particular problem, we will focus on the trial-and-error method, which involves systematically testing different combinations of factors until we find the correct pair. This method relies on understanding the relationship between the coefficients of the quadratic expression and the constants within the binomial factors. By carefully considering the signs and magnitudes of the coefficients and constants, we can efficiently narrow down the possibilities and identify the correct factors. This approach not only solves the problem at hand but also reinforces the fundamental principles of factoring quadratic expressions.
Method: Factoring by Inspection and Verification
The most efficient approach to solve this problem is factoring by inspection, followed by verification. This method involves identifying two numbers that satisfy specific conditions related to the coefficients of the quadratic expression. In our case, we need to find two numbers that:
- Multiply to the constant term (48).
- Add up to the coefficient of the linear term (-14).
Let's systematically analyze the factors of 48. The pairs of factors of 48 are (1, 48), (2, 24), (3, 16), (4, 12), and (6, 8). Since the coefficient of the linear term is negative (-14), and the constant term is positive (48), we know that both factors must be negative. This is because the product of two negative numbers is positive, and the sum of two negative numbers is negative. By considering the negative pairs, we can narrow down the possibilities and identify the pair that adds up to -14. This process requires a systematic approach and a keen eye for patterns.
Now, let's consider the negative pairs of factors: (-1, -48), (-2, -24), (-3, -16), (-4, -12), and (-6, -8). We need to find the pair that adds up to -14. By examining these pairs, we can quickly identify that -6 and -8 satisfy this condition: -6 + (-8) = -14. This means that the factors of the quadratic expression m^2 - 14m + 48 are likely to involve -6 and -8. To confirm this, we can construct the binomial factors using these numbers and then expand them to see if they match the original quadratic expression. This verification step is crucial to ensure that we have identified the correct factors.
Once we have identified the potential factors, we need to verify our solution by expanding the binomials and comparing the result to the original quadratic expression. This step is crucial to ensure that we have not made any mistakes in our factoring process. Expanding the binomials involves multiplying each term in the first binomial by each term in the second binomial and then simplifying the resulting expression. If the expanded expression matches the original quadratic expression, then we have successfully factored the expression. If not, we need to re-evaluate our factors and try a different combination. This verification process reinforces the relationship between factoring and expansion and helps to develop a deeper understanding of quadratic expressions.
Solution: Determining the Correct Factors
Based on our analysis, the numbers -6 and -8 satisfy the conditions for factoring the expression. Therefore, the factors are (m - 6) and (m - 8). To confirm this, let's expand the product of these binomials:
Since the expanded expression matches the original quadratic expression, we have successfully factored it. Now, let's compare our result with the given options:
- A) $(m-12)(m-4)$
- B) $(m-12)(m+4)$
- C) $(m-6)(m-8)$
- D) $(m+6)(m+8)$
By comparing our factored expression with the options, we can clearly see that option C, $(m-6)(m-8)$, is the correct answer. This confirms our analysis and demonstrates the effectiveness of the factoring by inspection method. The ability to quickly identify factors and verify them through expansion is a valuable skill in algebra and beyond. This skill allows us to solve equations, simplify expressions, and tackle more complex mathematical problems with confidence.
The process of expanding the other options can further illustrate why they are incorrect. For example, expanding option A, $(m-12)(m-4)$, gives us m^2 - 16m + 48, which does not match the original quadratic expression. Similarly, expanding option B, $(m-12)(m+4)$, gives us m^2 - 8m - 48, and expanding option D, $(m+6)(m+8)$, gives us m^2 + 14m + 48. None of these expansions match the original expression, further solidifying our conclusion that option C is the correct answer. This exercise highlights the importance of verification in the factoring process and reinforces the understanding of how binomial factors combine to form quadratic expressions.
Conclusion: Option C is the Correct Answer
Therefore, the factors of m^2 - 14m + 48 are (m - 6)(m - 8), which corresponds to option C. This solution demonstrates the application of factoring techniques to quadratic expressions. By systematically analyzing the coefficients and applying the principles of factoring, we successfully identified the correct factors. This problem serves as a valuable exercise in mastering factoring skills, which are essential for success in algebra and higher-level mathematics. The ability to factor quadratic expressions efficiently and accurately is a key skill for solving equations, graphing functions, and tackling various mathematical problems.
Factoring quadratic expressions is not just a mathematical exercise; it's a fundamental tool for solving real-world problems. Quadratic equations, which are closely related to quadratic expressions, appear in various fields, including physics, engineering, economics, and computer science. For example, quadratic equations can be used to model the trajectory of a projectile, the optimal design of a bridge, or the growth of a population. By mastering factoring techniques, you equip yourself with the ability to solve these types of problems and make informed decisions in various contexts. The applications of factoring extend far beyond the classroom, making it a valuable skill for personal and professional growth.
In conclusion, the correct factors of the quadratic expression m^2 - 14m + 48 are (m - 6)(m - 8). This solution was obtained by systematically analyzing the coefficients of the quadratic expression, identifying the appropriate factors, and verifying the result through expansion. This problem demonstrates the importance of factoring skills in algebra and highlights the versatility of quadratic expressions in various mathematical and real-world applications. By mastering factoring techniques, you gain a powerful tool for solving problems, making informed decisions, and advancing your understanding of mathematics and its applications.
