Solving M^2 - 12m + 20 = 0 By Factorization A Step-by-Step Guide
Introduction
In the realm of mathematics, solving quadratic equations is a fundamental skill. Quadratic equations, characterized by their highest power of the variable being two, appear in various applications, ranging from physics to engineering. One common method to tackle these equations is factorization. Factorization involves expressing the quadratic expression as a product of two linear factors. This article delves into the process of solving the quadratic equation m^2 - 12m + 20 = 0 by employing the factorization technique. We will break down each step, making it easy to understand even for those new to quadratic equations. By mastering this method, you'll gain a valuable tool for solving a wide range of mathematical problems.
Understanding Quadratic Equations
Before diving into the solution, it's crucial to grasp the basics of quadratic equations. A quadratic equation is generally represented in the form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The term 'a' cannot be zero, as that would make it a linear equation. The solutions to a quadratic equation are also known as its roots. These roots represent the values of 'x' that satisfy the equation. A quadratic equation can have two, one, or no real roots, depending on the discriminant (b^2 - 4ac). Understanding the nature of roots helps in choosing the appropriate method to solve the equation. In our case, we have m^2 - 12m + 20 = 0, where a = 1, b = -12, and c = 20. The discriminant will help determine if factorization is a viable approach. This preliminary understanding is crucial before we delve into the actual factorization process.
The Factorization Method: A Step-by-Step Guide
Factorization is a powerful technique to solve quadratic equations, especially when the roots are rational numbers. The core idea is to express the quadratic expression as a product of two linear factors. For the equation m^2 - 12m + 20 = 0, we aim to rewrite it in the form (m + p)(m + q) = 0, where 'p' and 'q' are constants. The process involves several key steps: First, identify the coefficients 'a', 'b', and 'c'. In our case, a = 1, b = -12, and c = 20. Next, find two numbers whose product equals 'ac' (1 * 20 = 20) and whose sum equals 'b' (-12). This is often the most challenging step, requiring some trial and error. Once you find these numbers, rewrite the middle term (-12m) using these numbers. This step transforms the quadratic expression into a four-term expression, which can then be factored by grouping. Finally, set each factor equal to zero and solve for 'm' to find the roots of the equation. This systematic approach ensures that the equation is solved accurately and efficiently. The success of the factorization method hinges on finding the correct pair of numbers that satisfy the product and sum conditions.
Step 1: Identify Coefficients
The first step in solving the quadratic equation m^2 - 12m + 20 = 0 by factorization is to identify the coefficients. In the standard form of a quadratic equation, ax^2 + bx + c = 0, 'a' is the coefficient of the squared term, 'b' is the coefficient of the linear term, and 'c' is the constant term. In our equation, m^2 - 12m + 20 = 0, the coefficient of m^2 (a) is 1, the coefficient of m (b) is -12, and the constant term (c) is 20. Correctly identifying these coefficients is crucial because they are used in subsequent steps of the factorization process. The values of 'a', 'b', and 'c' determine the numbers we need to find for factoring the quadratic expression. Misidentification of these coefficients will lead to incorrect factorization and ultimately incorrect solutions. Therefore, taking the time to accurately identify these values is a fundamental step in solving quadratic equations by factorization.
Step 2: Find Two Numbers
The second crucial step in solving the quadratic equation m^2 - 12m + 20 = 0 by factorization is to find two numbers that meet specific criteria related to the coefficients identified in the previous step. These two numbers, let's call them p and q, must satisfy two conditions: Their product (p * q) should be equal to the product of 'a' and 'c' (a * c), and their sum (p + q) should be equal to 'b'. In our case, a = 1, b = -12, and c = 20. Therefore, we need to find two numbers such that their product is 1 * 20 = 20 and their sum is -12. This often involves some trial and error, but systematically listing factor pairs of 20 can help. The factor pairs of 20 are (1, 20), (2, 10), (4, 5), and their negative counterparts. After considering these pairs, we find that -2 and -10 satisfy the conditions: (-2) * (-10) = 20 and (-2) + (-10) = -12. This step is the cornerstone of the factorization method, as finding these two numbers allows us to rewrite the middle term and proceed with factoring by grouping. The ability to identify these numbers efficiently is a skill honed with practice.
Step 3: Rewrite the Middle Term
Once we've identified the two numbers, -2 and -10, that satisfy the product and sum conditions, the next step in solving m^2 - 12m + 20 = 0 by factorization is to rewrite the middle term (-12m). This involves expressing -12m as the sum of the terms using the two numbers we found. Specifically, we replace -12m with -2m - 10m. This transformation does not change the value of the expression but allows us to factor by grouping in the subsequent step. The original equation m^2 - 12m + 20 = 0 now becomes m^2 - 2m - 10m + 20 = 0. This step is crucial because it sets up the expression for the next phase of factorization, where we will group terms and factor out common factors. Rewriting the middle term correctly is essential for the success of the factorization method, as it bridges the gap between the original quadratic expression and its factored form. Careful attention to signs and coefficients is important during this step to avoid errors.
Step 4: Factor by Grouping
After rewriting the middle term, the next step in solving m^2 - 12m + 20 = 0 by factorization is to factor by grouping. We now have the expression m^2 - 2m - 10m + 20 = 0. The process of factoring by grouping involves pairing the terms and factoring out the greatest common factor (GCF) from each pair. First, we group the first two terms and the last two terms: (m^2 - 2m) + (-10m + 20). Then, we factor out the GCF from each group. From the first group (m^2 - 2m), the GCF is 'm', so we factor it out to get m(m - 2). From the second group (-10m + 20), the GCF is -10, so we factor it out to get -10(m - 2). Now, the expression looks like this: m(m - 2) - 10(m - 2) = 0. Notice that (m - 2) is a common factor in both terms. We can factor out (m - 2) from the entire expression, resulting in (m - 2)(m - 10) = 0. This step is critical as it transforms the four-term expression into a product of two binomial factors. Factoring by grouping simplifies the equation, making it easier to find the solutions.
Step 5: Solve for m
Having successfully factored the quadratic expression, the final step in solving m^2 - 12m + 20 = 0 is to solve for m. We have the factored equation (m - 2)(m - 10) = 0. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This means either (m - 2) = 0 or (m - 10) = 0. Solving each of these linear equations gives us the values of 'm' that satisfy the original quadratic equation. For (m - 2) = 0, we add 2 to both sides to get m = 2. For (m - 10) = 0, we add 10 to both sides to get m = 10. Therefore, the solutions to the quadratic equation m^2 - 12m + 20 = 0 are m = 2 and m = 10. These values are the roots of the equation, meaning that if we substitute either 2 or 10 for 'm' in the original equation, it will hold true. This final step completes the factorization method, providing the solutions to the quadratic equation.
Alternative Methods for Solving Quadratic Equations
While factorization is an efficient method for solving quadratic equations, it's not always the most suitable approach. When factorization proves challenging or impossible, alternative methods come into play. Two prominent alternatives are the quadratic formula and completing the square. The quadratic formula is a universal method that can solve any quadratic equation, regardless of whether it can be factored. It provides the roots directly using the coefficients 'a', 'b', and 'c' from the standard form equation. Completing the square, on the other hand, involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. Each method has its strengths and weaknesses, and the choice depends on the specific equation and the solver's preference. Understanding these alternative methods broadens the problem-solving toolkit and allows for a more flexible approach to tackling quadratic equations.
The Quadratic Formula
The quadratic formula is a versatile tool for solving quadratic equations of the form ax^2 + bx + c = 0. Unlike factorization, which relies on finding suitable factors, the quadratic formula provides a direct solution regardless of the equation's factorability. The formula is given by: x = [-b ± √(b^2 - 4ac)] / (2a). To apply the formula, one must first identify the coefficients 'a', 'b', and 'c' from the quadratic equation. Then, these values are substituted into the formula, and the expression is simplified to find the roots. The discriminant (b^2 - 4ac) within the formula determines the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are two complex roots. In the case of m^2 - 12m + 20 = 0, a = 1, b = -12, and c = 20. Substituting these values into the quadratic formula yields the solutions m = 2 and m = 10, consistent with the factorization method. The quadratic formula is particularly useful when the roots are irrational or complex, as factorization may not be feasible in such cases.
Completing the Square
Completing the square is another powerful method for solving quadratic equations, providing a systematic way to transform the equation into a form where the variable can be easily isolated. The method involves manipulating the equation ax^2 + bx + c = 0 to create a perfect square trinomial on one side. This is achieved by adding and subtracting a specific value, (b/2a)^2, to the equation. The process typically involves several steps. First, divide the equation by 'a' if 'a' is not equal to 1. Then, move the constant term to the right side of the equation. Next, add (b/2a)^2 to both sides to complete the square. This transforms the left side into a perfect square trinomial, which can be written as (x + b/2a)^2. Finally, take the square root of both sides and solve for 'x'. For the equation m^2 - 12m + 20 = 0, completing the square involves adding and subtracting (12/2)^2 = 36. This leads to (m - 6)^2 = 16, from which we find m = 2 and m = 10. Completing the square is not only a method for solving quadratic equations but also a technique used in various mathematical contexts, such as finding the vertex of a parabola or deriving the quadratic formula itself. While it may involve more steps than the quadratic formula, it provides valuable insights into the structure of quadratic equations.
Conclusion
In conclusion, solving quadratic equations by factorization is a fundamental skill in mathematics. Through a step-by-step process, we've demonstrated how to factorize the equation m^2 - 12m + 20 = 0, arriving at the solutions m = 2 and m = 10. We began by identifying the coefficients, then found two numbers whose product and sum satisfied specific conditions, rewrote the middle term, factored by grouping, and finally, solved for 'm'. While factorization is an effective method, we also explored alternative approaches like the quadratic formula and completing the square, which offer solutions for a broader range of quadratic equations. Mastering these techniques equips one with a versatile toolkit for tackling various mathematical problems involving quadratic expressions. Understanding the nuances of each method allows for a strategic approach to problem-solving, ensuring accuracy and efficiency. Whether it's factorization, the quadratic formula, or completing the square, the ability to solve quadratic equations is a cornerstone of mathematical proficiency.
