Solving 2n² - 15n + 7 = 0 By Factorization A Step-by-Step Guide
Introduction
In this comprehensive guide, we will delve into the method of solving quadratic equations through factorization. Specifically, we will address the equation 2n² - 15n + 7 = 0. This method is a fundamental technique in algebra, offering a clear and intuitive approach to finding the roots of quadratic equations. Understanding factorization is crucial not only for solving equations but also for grasping more advanced concepts in mathematics and related fields. In this guide, we will break down the process step-by-step, ensuring a thorough understanding of each stage. Whether you are a student grappling with quadratic equations for the first time or someone looking to refresh your algebra skills, this guide provides a detailed and accessible explanation. We'll explore the underlying principles, demonstrate the factorization process, and discuss the significance of the solutions obtained. By the end of this guide, you will be well-equipped to tackle similar quadratic equations with confidence and precision.
Understanding Quadratic Equations
Before diving into the specifics of solving 2n² - 15n + 7 = 0, it's essential to understand the nature of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. The coefficients a, b, and c play a crucial role in determining the characteristics and solutions of the equation. The coefficient a determines the shape of the parabola when the quadratic equation is graphed, and b and c influence its position and intercepts.
The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These roots represent the points where the parabola intersects the x-axis on a graph. A quadratic equation can have two distinct real roots, one repeated real root, or two complex roots. The nature of the roots depends on the discriminant, which is given by the formula Δ = b² - 4ac. If Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; and if Δ < 0, there are two complex roots. Recognizing the structure and properties of quadratic equations is the first step in effectively solving them, whether by factorization, completing the square, or using the quadratic formula. Each method offers a unique approach, and choosing the most suitable one often depends on the specific equation and the ease of manipulation.
The Factorization Method: A Step-by-Step Guide
The factorization method is a powerful technique for solving quadratic equations, especially when the equation can be easily factored. This method involves expressing the quadratic expression as a product of two linear factors. For the given equation, 2n² - 15n + 7 = 0, our goal is to find two binomials that, when multiplied together, yield the original quadratic expression. The general strategy involves finding two numbers that satisfy specific conditions related to the coefficients of the quadratic equation.
Specifically, we need to find two numbers that multiply to the product of the leading coefficient (a) and the constant term (c), and add up to the middle coefficient (b). In our case, a = 2, b = -15, and c = 7. Thus, we are looking for two numbers that multiply to (2 * 7) = 14 and add up to -15. After some consideration, we find that the numbers -1 and -14 satisfy these conditions. The next step is to rewrite the middle term (-15n) using these two numbers. This process is known as splitting the middle term. We rewrite -15n as -1n - 14n, so the equation becomes 2n² - 1n - 14n + 7 = 0. Now, we factor by grouping. We group the first two terms and the last two terms together: (2n² - 1n) + (-14n + 7) = 0. We factor out the greatest common factor (GCF) from each group. From the first group, the GCF is n, and from the second group, the GCF is -7. This gives us n(2n - 1) - 7(2n - 1) = 0. Notice that (2n - 1) is a common factor in both terms. We factor out (2n - 1) to get (2n - 1)(n - 7) = 0. Finally, we set each factor equal to zero and solve for n. This gives us two linear equations: 2n - 1 = 0 and n - 7 = 0. Solving these equations yields the solutions for n.
Applying Factorization to 2n² - 15n + 7 = 0
To solve the equation 2n² - 15n + 7 = 0 by factorization, we follow the step-by-step method outlined earlier. The first step is to identify the coefficients a, b, and c in the quadratic equation. In this case, a = 2, b = -15, and c = 7. Next, we need to find two numbers that multiply to the product of a and c (which is 2 * 7 = 14) and add up to b (which is -15). As determined earlier, these two numbers are -1 and -14 because (-1) * (-14) = 14 and (-1) + (-14) = -15. Now, we rewrite the middle term (-15n) using these two numbers. This gives us 2n² - 1n - 14n + 7 = 0. The next step is to factor by grouping. We group the first two terms and the last two terms together: (2n² - 1n) + (-14n + 7) = 0. We factor out the greatest common factor (GCF) from each group. From the first group (2n² - 1n), the GCF is n, and from the second group (-14n + 7), the GCF is -7. Factoring out the GCFs, we get n(2n - 1) - 7(2n - 1) = 0. We observe that (2n - 1) is a common factor in both terms. We factor out (2n - 1) to obtain (2n - 1)(n - 7) = 0. The final step is to set each factor equal to zero and solve for n. This gives us two linear equations: 2n - 1 = 0 and n - 7 = 0. Solving the first equation, 2n - 1 = 0, we add 1 to both sides to get 2n = 1, and then divide by 2 to get n = 1/2. Solving the second equation, n - 7 = 0, we add 7 to both sides to get n = 7. Thus, the solutions to the quadratic equation 2n² - 15n + 7 = 0 are n = 1/2 and n = 7.
Solutions and Verification
After solving the equation 2n² - 15n + 7 = 0 by factorization, we have found two solutions: n = 1/2 and n = 7. It is crucial to verify these solutions to ensure they satisfy the original equation. Verification involves substituting each solution back into the original equation and confirming that the equation holds true. First, let's verify n = 1/2. Substituting n = 1/2 into the equation 2n² - 15n + 7 = 0, we get: 2(1/2)² - 15(1/2) + 7 = 0 2(1/4) - 15/2 + 7 = 0 1/2 - 15/2 + 7 = 0 To simplify, we can combine the fractions: (1 - 15)/2 + 7 = 0 -14/2 + 7 = 0 -7 + 7 = 0 0 = 0 The equation holds true for n = 1/2, so it is a valid solution. Now, let's verify n = 7. Substituting n = 7 into the equation 2n² - 15n + 7 = 0, we get: 2(7)² - 15(7) + 7 = 0 2(49) - 105 + 7 = 0 98 - 105 + 7 = 0 -7 + 7 = 0 0 = 0 The equation also holds true for n = 7, confirming that it is a valid solution. Therefore, the solutions n = 1/2 and n = 7 are correct for the quadratic equation 2n² - 15n + 7 = 0. This verification process is an essential step in solving any equation, as it ensures the accuracy of the solutions and helps to avoid errors.
Alternative Methods for Solving Quadratic Equations
While factorization is a powerful method for solving quadratic equations, it is not always the most straightforward or applicable technique. There are alternative methods available, each with its own advantages and disadvantages. Two prominent methods are the quadratic formula and completing the square. The quadratic formula is a universal method that can be used to solve any quadratic equation, regardless of whether it can be factored easily. The formula is given by: n = (-b ± √(b² - 4ac)) / (2a) where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. This formula is derived from the process of completing the square and provides a direct way to find the solutions without needing to factor. Applying the quadratic formula to the equation 2n² - 15n + 7 = 0, we have a = 2, b = -15, and c = 7. Substituting these values into the formula, we get: n = (15 ± √((-15)² - 4(2)(7))) / (2(2)) n = (15 ± √(225 - 56)) / 4 n = (15 ± √169) / 4 n = (15 ± 13) / 4 This gives us two possible solutions: n = (15 + 13) / 4 = 28 / 4 = 7 n = (15 - 13) / 4 = 2 / 4 = 1/2 These solutions match the ones we found by factorization, confirming the accuracy of both methods. Another method is completing the square, which involves transforming the quadratic equation into a perfect square trinomial. This method is particularly useful for deriving the quadratic formula and for understanding the structure of quadratic equations. However, it can be more complex and time-consuming than factorization or using the quadratic formula, especially when the coefficients are not simple integers. The choice of method depends on the specific equation and the solver's preference and familiarity with each technique.
Conclusion
In conclusion, we have successfully solved the quadratic equation 2n² - 15n + 7 = 0 using the method of factorization. This method involved identifying the coefficients, finding two numbers that multiply to ac and add up to b, rewriting the middle term, factoring by grouping, and setting each factor equal to zero to find the solutions. The solutions obtained were n = 1/2 and n = 7, which were then verified by substituting them back into the original equation. We also explored alternative methods for solving quadratic equations, such as the quadratic formula and completing the square. The quadratic formula provides a direct and universal approach, while completing the square offers insights into the structure of quadratic equations but can be more complex. The choice of method often depends on the specific equation and the solver's preference. Understanding these different methods enhances one's ability to tackle a wide range of quadratic equations effectively. Mastering the technique of factorization is a fundamental skill in algebra, providing a solid foundation for more advanced mathematical concepts. By following the step-by-step approach outlined in this guide, students and math enthusiasts can confidently solve quadratic equations and deepen their understanding of algebraic principles.
