Solving 6u² - 37u + 6 = 0 A Step-by-Step Guide
In the realm of mathematics, quadratic equations hold a significant position. These equations, characterized by their highest power of 2, frequently appear in various fields, from physics and engineering to economics and computer science. Understanding how to solve quadratic equations is, therefore, a fundamental skill for anyone delving into these disciplines. This comprehensive guide will walk you through the process of solving the quadratic equation 6u² - 37u + 6 = 0, demonstrating a crucial technique applicable to a wide range of similar problems. We will explore the process step-by-step, ensuring a clear and thorough understanding of the methodology. Mastering the techniques for solving quadratic equations opens doors to tackling more complex mathematical problems and real-world applications. This article provides not only a solution but also an in-depth exploration of the underlying principles, making it a valuable resource for students, educators, and anyone interested in enhancing their mathematical prowess. Understanding the nuances of solving such equations helps in developing critical thinking and problem-solving skills that extend beyond the realm of mathematics.
Understanding Quadratic Equations
Before diving into the solution, let's first understand the general form of a quadratic equation: ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Our given equation, 6u² - 37u + 6 = 0, perfectly fits this form, with a = 6, b = -37, and c = 6. The goal is to find the values of 'u' that satisfy this equation, often referred to as the roots or solutions of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages and is suitable for different types of quadratic equations. Factoring, for instance, is often the quickest method when the equation can be easily factored. Completing the square is a more general method that can be used to solve any quadratic equation. The quadratic formula is a direct application of the completing the square method and provides a straightforward way to find the solutions. The choice of method often depends on the specific equation and personal preference. A deep understanding of each method allows for flexibility and efficiency in problem-solving. Moreover, recognizing the structure of the quadratic equation and its coefficients can provide insights into the nature of the solutions, such as whether they are real or complex, rational or irrational. This foundational knowledge is crucial for advanced mathematical studies and practical applications.
Method 1: Factoring the Quadratic Equation
Factoring is a powerful technique for solving quadratic equations, particularly when the equation can be expressed as a product of two binomials. In the case of 6u² - 37u + 6 = 0, our objective is to find two binomials that, when multiplied, yield the original quadratic expression. This involves identifying two numbers that multiply to give the product of the leading coefficient (6) and the constant term (6), which is 36, and add up to the middle coefficient (-37). The numbers -36 and -1 satisfy these conditions. We can rewrite the middle term (-37u) as -36u - u, which allows us to factor by grouping. This process involves splitting the middle term into two terms that enable us to factor out common factors from pairs of terms. By rewriting the equation, we transform it into a form that is easier to manipulate and solve. The key to successful factoring lies in recognizing the relationships between the coefficients and finding the appropriate factors. This skill is honed through practice and familiarity with different types of quadratic expressions. Factoring not only provides a solution to the equation but also enhances understanding of the underlying algebraic structure. It serves as a fundamental building block for more advanced mathematical concepts and techniques. Furthermore, the ability to factor efficiently is a valuable asset in various mathematical and scientific contexts, making it a core skill for anyone pursuing studies in these fields.
Step-by-Step Factoring
Let's proceed with the factoring process. We rewrite the equation 6u² - 37u + 6 = 0 as 6u² - 36u - u + 6 = 0. Now, we can factor by grouping. From the first two terms, 6u² - 36u, we can factor out 6u, which gives us 6u(u - 6). From the last two terms, -u + 6, we can factor out -1, resulting in -1(u - 6). Now, we have 6u(u - 6) - 1(u - 6) = 0. Notice that (u - 6) is a common factor, so we can factor it out, leading to (6u - 1)(u - 6) = 0. This factored form is a significant step, as it transforms the quadratic equation into a product of two linear factors. Each factor represents a potential solution to the equation. The next step involves setting each factor equal to zero and solving for 'u'. This is based on the principle that if the product of two factors is zero, then at least one of the factors must be zero. By applying this principle, we can find the values of 'u' that make the equation true. This method is not only efficient but also provides a clear understanding of the relationship between the roots of the equation and its factors. Moreover, the factored form can be used to analyze the behavior of the quadratic function and its graph, providing insights into its intercepts and vertex.
Solving for u
Setting each factor to zero, we have two equations: 6u - 1 = 0 and u - 6 = 0. Solving the first equation, 6u - 1 = 0, we add 1 to both sides, giving us 6u = 1. Then, we divide both sides by 6, resulting in u = 1/6. Solving the second equation, u - 6 = 0, we add 6 to both sides, which gives us u = 6. Thus, we have found two solutions for the quadratic equation 6u² - 37u + 6 = 0. These solutions, u = 1/6 and u = 6, are the values of 'u' that make the equation true. They represent the points where the parabola defined by the quadratic equation intersects the x-axis. The solutions are expressed in the simplest form, as requested. Understanding how to derive these solutions is crucial for mastering quadratic equations. The process involves careful algebraic manipulation and a clear understanding of the properties of equations. Moreover, verifying the solutions by substituting them back into the original equation is a good practice to ensure accuracy. This step-by-step approach not only provides the solutions but also reinforces the fundamental principles of equation solving. Furthermore, the ability to solve quadratic equations is a cornerstone of algebra and is essential for tackling more advanced mathematical problems.
Method 2: Using the Quadratic Formula
When factoring proves challenging or time-consuming, the quadratic formula provides a reliable alternative for solving quadratic equations. The quadratic formula is a universal tool that can be applied to any quadratic equation in the form ax² + bx + c = 0. The formula is given by: u = (-b ± √(b² - 4ac)) / (2a). This formula is derived from the method of completing the square and provides a direct way to find the solutions of the quadratic equation. It is particularly useful when the roots are irrational or complex, as it does not rely on factoring. The quadratic formula is a fundamental concept in algebra and is widely used in various mathematical and scientific fields. Its application requires careful substitution of the coefficients a, b, and c into the formula and subsequent simplification. The discriminant, b² - 4ac, plays a crucial role in determining the nature of the solutions. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution (a repeated root); and if it is negative, there are two complex solutions. Understanding the quadratic formula and its applications is essential for mastering quadratic equations and their related concepts. Moreover, the ability to apply the formula accurately and efficiently is a valuable skill in problem-solving and mathematical analysis.
Applying the Formula
In our equation, 6u² - 37u + 6 = 0, we have a = 6, b = -37, and c = 6. Substituting these values into the quadratic formula, we get: u = (-(-37) ± √((-37)² - 4 * 6 * 6)) / (2 * 6). This substitution is a critical step, as any error in this process will lead to incorrect solutions. Careful attention to detail is essential when substituting the values, especially the signs. The next step involves simplifying the expression under the square root and performing the arithmetic operations. This requires a solid understanding of the order of operations and the properties of square roots. The expression under the square root, known as the discriminant, provides valuable information about the nature of the solutions. In this case, the discriminant is positive, indicating that there are two distinct real solutions. Simplifying the expression further, we can find the exact values of these solutions. The quadratic formula not only provides a solution but also offers insights into the structure and behavior of quadratic equations. Its versatility and reliability make it an indispensable tool in mathematics and its applications. Furthermore, mastering the quadratic formula is a fundamental step towards tackling more advanced mathematical concepts and problems.
Simplifying the Solution
Simplifying further, we have u = (37 ± √(1369 - 144)) / 12, which simplifies to u = (37 ± √1225) / 12. Since √1225 = 35, we get u = (37 ± 35) / 12. This gives us two possible solutions: u = (37 + 35) / 12 and u = (37 - 35) / 12. For u = (37 + 35) / 12, we have u = 72 / 12, which simplifies to u = 6. For u = (37 - 35) / 12, we have u = 2 / 12, which simplifies to u = 1/6. Therefore, the solutions are u = 6 and u = 1/6, which match the solutions we found by factoring. This consistency between the two methods provides a strong confirmation of the accuracy of our solutions. Simplifying the solutions to their simplest form is an important step, as it ensures clarity and ease of interpretation. The ability to simplify expressions involving square roots and fractions is a fundamental skill in algebra. Moreover, comparing the solutions obtained by different methods is a valuable practice for verifying results and enhancing understanding. The quadratic formula, with its systematic approach, offers a robust method for solving quadratic equations, especially when factoring is not straightforward. Its application reinforces the importance of algebraic manipulation and the properties of mathematical operations.
Conclusion: Solutions and Implications
In conclusion, by both factoring and using the quadratic formula, we have found the solutions to the equation 6u² - 37u + 6 = 0 to be u = 6 and u = 1/6. These solutions are expressed as an integer and a proper fraction, respectively, in their simplest forms. The fact that we arrived at the same solutions using two different methods underscores the correctness of our results. The solutions represent the values of 'u' that satisfy the given quadratic equation. They are also the x-intercepts of the parabola represented by the equation y = 6u² - 37u + 6. Understanding these solutions is crucial for various applications in mathematics and other fields. For instance, in physics, these solutions might represent the time at which an object reaches a certain height when its motion is described by a quadratic equation. In economics, they could represent the equilibrium points in a supply and demand model. The ability to solve quadratic equations is a fundamental skill that opens doors to a wide range of problem-solving opportunities. Moreover, the process of solving these equations reinforces critical thinking and analytical skills. The techniques discussed in this guide, including factoring and using the quadratic formula, provide a comprehensive toolkit for tackling quadratic equations of various forms and complexities. These skills are essential for anyone pursuing studies in mathematics, science, engineering, and related fields.
The solutions to the equation 6u² - 37u + 6 = 0 are:
