Solving M^2-12m+20=0 By Factorisation A Step-by-Step Guide
Understanding quadratic equations is a fundamental skill in algebra, serving as a cornerstone for more advanced mathematical concepts. Quadratic equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, frequently appear in various scientific and engineering applications. Mastering the techniques to solve these equations is therefore crucial. Among the several methods available, factorisation stands out as an elegant and efficient approach, particularly when the equation can be expressed as a product of simple factors. This method not only provides the solutions, also known as roots, of the equation but also enhances understanding of the equation's structure. In this article, we delve into the specifics of solving the quadratic equation m² - 12m + 20 = 0 by factorisation, offering a step-by-step guide to make the process clear and accessible. Whether you're a student grappling with algebra or someone looking to refresh your mathematical skills, this comprehensive explanation will help you confidently tackle similar problems. The factorisation method hinges on rewriting the quadratic equation as a product of two binomial factors. This transformation allows us to utilize the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. The challenge lies in identifying the correct factors, which requires a systematic approach and a keen eye for numerical relationships. By breaking down the steps and providing clear examples, we aim to demystify the factorisation process and equip you with the skills to solve quadratic equations effectively.
Factorising Quadratic Equations: The Basics
Before we tackle the specific equation m² - 12m + 20 = 0, let's solidify the foundational principles of factorising quadratic equations. Factorisation is essentially the reverse process of expansion. When we expand (x + a)(x + b), we get x² + (a + b)x + ab. Conversely, factorising a quadratic equation involves transforming it from the form ax² + bx + c back into the product of two binomials. This transformation is particularly useful for solving equations because it allows us to apply the zero-product property, which states that if AB = 0, then either A = 0 or B = 0 (or both). The key to successful factorisation lies in finding two numbers that satisfy specific conditions related to the coefficients of the quadratic equation. For a quadratic equation in the form x² + bx + c = 0, we need to find two numbers that add up to b and multiply to c. These numbers will then form the constant terms in our binomial factors. When dealing with quadratic equations where the leading coefficient (a) is not 1, the process becomes slightly more complex but still follows the same fundamental principles. We need to consider not only the factors of the constant term (c) but also the factors of the leading coefficient (a). This often involves a bit more trial and error, but with practice, you can develop an intuition for quickly identifying the correct factors. It's also important to note that not all quadratic equations can be easily factorised using integers. In such cases, other methods like the quadratic formula or completing the square may be more appropriate. However, when factorisation is possible, it often provides the most straightforward and efficient path to finding the solutions. Factorisation is not just a technique for solving equations; it is also a valuable tool for simplifying expressions and understanding the structure of algebraic relationships. By mastering factorisation, you gain a deeper appreciation for the connections between different areas of mathematics.
Step-by-Step Solution for m² - 12m + 20 = 0
Now, let's apply the principles of factorisation to solve the equation m² - 12m + 20 = 0. This equation is a classic example of a quadratic equation that can be readily solved by factorisation. The first step is to identify the coefficients. In this case, we have a = 1, b = -12, and c = 20. Our goal is to find two numbers that add up to b (-12) and multiply to c (20). A systematic approach is to list the pairs of factors of c (20) and see which pair, when added, gives us b (-12). The factors of 20 are: 1 and 20, 2 and 10, 4 and 5. Since we need the numbers to add up to -12, we should consider the negative factors as well. So, we also have -1 and -20, -2 and -10, -4 and -5. Looking at these pairs, we can see that -2 and -10 satisfy our conditions. They add up to -12 (-2 + -10 = -12) and multiply to 20 (-2 * -10 = 20). Now that we have our two numbers, we can rewrite the quadratic equation in factored form. The factored form will be (m - 2)(m - 10) = 0. This is because (m - 2)(m - 10) expands to m² - 10m - 2m + 20, which simplifies to m² - 12m + 20. The next step is to apply the zero-product property. This means that either (m - 2) = 0 or (m - 10) = 0. Solving each of these equations separately gives us the solutions for m. For m - 2 = 0, we add 2 to both sides, which gives us m = 2. For m - 10 = 0, we add 10 to both sides, which gives us m = 10. Therefore, the solutions to the quadratic equation m² - 12m + 20 = 0 are m = 2 and m = 10. We can verify these solutions by substituting them back into the original equation. If we substitute m = 2, we get 2² - 12(2) + 20 = 4 - 24 + 20 = 0, which is true. If we substitute m = 10, we get 10² - 12(10) + 20 = 100 - 120 + 20 = 0, which is also true. This confirms that our solutions are correct. By following these steps, you can confidently solve quadratic equations by factorisation. Remember to always check your solutions by substituting them back into the original equation.
Common Mistakes to Avoid When Factorising
When factorising quadratic equations, it's easy to make mistakes if you're not careful. Identifying and avoiding these common pitfalls can significantly improve your accuracy and understanding. One of the most frequent errors is incorrectly determining the signs of the factors. Remember that the signs of the constant terms in the binomial factors are crucial. If the constant term (c) in the quadratic equation is positive, the two numbers you're looking for must have the same sign (either both positive or both negative). If c is negative, the two numbers must have opposite signs. Furthermore, the sign of the linear term (b) dictates which sign should be dominant. For example, in m² - 12m + 20 = 0, the constant term (20) is positive, and the linear term (-12) is negative, indicating that both factors should be negative. Another common mistake is failing to consider all the factor pairs of the constant term. It's essential to systematically list out all possible pairs to ensure you don't miss the correct combination. For instance, when factorising x² + 5x + 6, some might overlook the factors 2 and 3, focusing only on 1 and 6. Rushing through the process can also lead to errors. Factorisation requires careful attention to detail, and skipping steps or making assumptions can result in incorrect factors. Always double-check your work and ensure that the factors you've identified actually multiply to give the original quadratic expression. A good way to verify your factorisation is to expand the binomial factors you've found and see if they match the original equation. For example, if you factor x² - 4x + 3 as (x - 1)(x - 2), expanding this gives x² - 3x + 2, which is not the original equation. The correct factorisation is (x - 1)(x - 3). Finally, it's important to remember that not all quadratic equations can be factorised using integers. In such cases, attempting to force a factorisation will only lead to frustration. If you've spent a reasonable amount of time trying to factor and haven't succeeded, it may be necessary to use alternative methods like the quadratic formula or completing the square. By being aware of these common mistakes and taking steps to avoid them, you can become more proficient and confident in factorising quadratic equations.
Real-World Applications of Quadratic Equations
Quadratic equations are not just abstract mathematical concepts; they have a wide range of real-world applications across various fields. Understanding and solving quadratic equations is essential for tackling problems in physics, engineering, economics, and even everyday scenarios. In physics, quadratic equations are fundamental in describing projectile motion. The trajectory of a projectile, such as a ball thrown into the air, can be modeled using a quadratic equation that takes into account factors like initial velocity, launch angle, and gravity. By solving the quadratic equation, physicists can determine the range, maximum height, and time of flight of the projectile. This knowledge is crucial in fields like sports, ballistics, and aerospace engineering. Engineers also heavily rely on quadratic equations in designing structures and systems. For example, in civil engineering, quadratic equations are used to calculate the optimal shape of arches and bridges to ensure structural stability. In electrical engineering, they are used to analyze circuits and determine the flow of current. The design of lenses and mirrors in optical systems also involves quadratic equations, as the focusing properties of these elements are described by parabolic curves, which are represented by quadratic functions. In economics, quadratic equations can be used to model cost, revenue, and profit functions. Businesses use these models to determine optimal pricing strategies and production levels that maximize profits. For instance, the break-even point, where total revenue equals total cost, can often be found by solving a quadratic equation. Quadratic equations also appear in computer graphics and game development. The rendering of curved surfaces and the simulation of realistic motion often involve quadratic functions. Understanding these equations is essential for creating visually appealing and interactive experiences. Even in everyday life, we encounter situations where quadratic equations are relevant. For example, if you're trying to calculate the area of a rectangular garden with a fixed perimeter, you might end up solving a quadratic equation to find the dimensions that maximize the area. Similarly, if you're planning a budget and need to account for compound interest, you'll be dealing with an equation that has a quadratic component. The ubiquity of quadratic equations underscores the importance of mastering the techniques for solving them. Whether you're a student, a professional, or simply someone who enjoys problem-solving, a solid understanding of quadratic equations will undoubtedly prove valuable.
Conclusion: Mastering Factorisation and Beyond
In conclusion, mastering factorisation is a crucial step in developing strong algebraic skills and solving quadratic equations efficiently. By understanding the principles behind factorisation and practicing consistently, you can confidently tackle a wide range of mathematical problems. The step-by-step solution of m² - 12m + 20 = 0 illustrates the power and elegance of this method. We've seen how to identify the coefficients, find the correct factors, and apply the zero-product property to arrive at the solutions. Furthermore, we've highlighted common mistakes to avoid and emphasized the importance of verifying your solutions to ensure accuracy. However, factorisation is just one tool in the toolbox for solving quadratic equations. While it is often the most straightforward method when applicable, not all quadratic equations can be easily factorised using integers. In such cases, alternative methods like the quadratic formula or completing the square become essential. The quadratic formula, in particular, provides a universal solution for any quadratic equation, regardless of whether it can be factorised. Completing the square is another powerful technique that not only solves quadratic equations but also provides insights into the vertex form of the quadratic function, which is useful for graphing and optimization problems. Beyond solving equations, understanding quadratic functions and their properties is vital for many applications. The graph of a quadratic function is a parabola, and the shape and characteristics of the parabola have significant implications in fields like physics, engineering, and economics. Whether you're modeling projectile motion, designing structures, or analyzing market trends, a deep understanding of quadratic functions will serve you well. As you continue your mathematical journey, remember that practice is key. The more you work with quadratic equations and factorisation, the more intuitive the process will become. Don't be afraid to make mistakes – they are valuable learning opportunities. Embrace the challenge, and you'll find that mastering factorisation and quadratic equations opens doors to a deeper appreciation of mathematics and its applications in the world around us.
