Solving $12+5x(x+6)=0$ Using The Quadratic Formula: A Step-by-Step Guide
Introduction: Mastering the Quadratic Formula
In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, quadratic equations hold a significant place due to their frequent appearance in diverse fields like physics, engineering, and economics. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. One of the most reliable methods for finding the solutions (also called roots) of a quadratic equation is the quadratic formula. This article delves into the application of the quadratic formula, providing a step-by-step guide to solving quadratic equations effectively. We will illustrate the process by solving the specific equation 12 + 5x(x + 6) = 0, ensuring a clear understanding of each step involved. By mastering the quadratic formula, you equip yourself with a powerful tool applicable across various mathematical and real-world scenarios. Understanding the quadratic formula is not just about memorizing a formula; it's about grasping a concept that underpins numerous mathematical applications. This guide will help you develop a strong foundation in solving quadratic equations, enhancing your problem-solving abilities in mathematics and beyond. Remember, practice is key to mastering any mathematical concept, so work through the example provided and try solving other quadratic equations on your own. The journey to mathematical proficiency is a continuous one, and mastering the quadratic formula is a significant milestone in that journey.
Understanding the Quadratic Formula: The Key to Solving Equations
The quadratic formula is a cornerstone of algebra, providing a universal method to solve quadratic equations. These equations, characterized by the form ax² + bx + c = 0, can often seem daunting. However, the quadratic formula elegantly provides the solutions, or roots, regardless of the complexity of the coefficients a, b, and c. The formula itself is expressed as: x = (-b ± √(b² - 4ac)) / 2a. Here, the symbol '±' indicates that there are typically two solutions, one obtained by adding the square root term and the other by subtracting it. The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the solutions. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has exactly one real solution (a repeated root). If the discriminant is negative, the equation has two complex solutions. This ability to predict the nature of the solutions before even applying the full formula highlights the power and elegance of the quadratic formula. Understanding the significance of the discriminant is essential for a complete grasp of quadratic equations. It allows you to anticipate the type of solutions you will encounter, guiding your problem-solving approach. The quadratic formula is not just a tool for finding answers; it's a gateway to deeper insights into the behavior of quadratic equations. By understanding the formula and its components, you can tackle a wide range of problems with confidence and precision. The versatility of the formula makes it an indispensable tool in various mathematical contexts, from simple textbook exercises to complex real-world applications. Remember, the key to mastering the quadratic formula lies in understanding its components and practicing its application.
Step-by-Step Solution: Applying the Quadratic Formula to 12 + 5x(x + 6) = 0
To effectively solve the equation 12 + 5x(x + 6) = 0 using the quadratic formula, we need to follow a structured approach. This involves transforming the equation into the standard quadratic form, identifying the coefficients, and then applying the formula. Let's break down the process step by step. First, we need to expand and simplify the given equation: 12 + 5x(x + 6) = 0. Expanding the term 5x(x + 6) gives us 5x² + 30x. So, the equation becomes 12 + 5x² + 30x = 0. Next, we rearrange the terms to match the standard quadratic form ax² + bx + c = 0. This gives us 5x² + 30x + 12 = 0. Now, we can clearly identify the coefficients: a = 5, b = 30, and c = 12. With the coefficients identified, we can substitute these values into the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. Plugging in the values, we get: x = (-30 ± √(30² - 4 * 5 * 12)) / (2 * 5). Now, we simplify the expression. First, calculate the discriminant: b² - 4ac = 30² - 4 * 5 * 12 = 900 - 240 = 660. So, the equation becomes: x = (-30 ± √660) / 10. We can simplify the square root by factoring out the largest perfect square from 660. 660 can be factored as 4 * 165, so √660 = √(4 * 165) = 2√165. Substituting this back into the equation, we get: x = (-30 ± 2√165) / 10. Finally, we simplify the expression by dividing both the numerator and denominator by 2: x = (-15 ± √165) / 5. This gives us two solutions: x = (-15 + √165) / 5 and x = (-15 - √165) / 5. These are the exact solutions to the given quadratic equation. By following these steps meticulously, we have successfully applied the quadratic formula to solve the equation. Each step is crucial in arriving at the correct solutions, and understanding the logic behind each step is essential for mastering this method.
Exact Solutions and Their Significance: Understanding the Roots
In the previous section, we meticulously applied the quadratic formula to solve the equation 12 + 5x(x + 6) = 0 and arrived at the exact solutions: x = (-15 + √165) / 5 and x = (-15 - √165) / 5. These solutions represent the roots of the quadratic equation, the values of x that make the equation true. It's important to understand the significance of these exact solutions. They provide the most accurate representation of the roots, without any rounding or approximation. This is particularly crucial in situations where precision is paramount, such as in engineering or scientific calculations. The presence of the square root term, √165, indicates that the solutions are irrational numbers. Irrational numbers cannot be expressed as a simple fraction and have a non-repeating, non-terminating decimal representation. Therefore, while we can approximate these solutions as decimals, the exact form provides the most accurate and complete answer. The two solutions also reflect the fundamental nature of quadratic equations, which typically have two roots. These roots may be real and distinct, real and repeated (in the case of a zero discriminant), or complex (in the case of a negative discriminant). In our example, the discriminant was positive, leading to two distinct real roots. Understanding the nature of the roots is crucial in interpreting the solutions in the context of the problem. For instance, in a physical application, the roots might represent points of intersection, equilibrium points, or other significant values. The exact solutions, therefore, provide a precise understanding of these critical points. In summary, the exact solutions obtained using the quadratic formula offer the most accurate and complete representation of the roots of a quadratic equation. Their significance lies in their precision and their ability to provide insights into the nature of the equation and its solutions.
Alternative Methods and Their Comparison: Exploring Beyond the Quadratic Formula
While the quadratic formula is a powerful and universally applicable method for solving quadratic equations, it's not the only approach. Other methods, such as factoring and completing the square, offer alternative pathways to finding solutions. Understanding these methods and their relative strengths and weaknesses can enhance your problem-solving toolkit. Factoring is a method that involves expressing the quadratic equation as a product of two linear factors. For example, the equation x² + 5x + 6 = 0 can be factored as (x + 2)(x + 3) = 0, leading to the solutions x = -2 and x = -3. Factoring is often the quickest method when it's applicable, but it's not always straightforward, especially when the coefficients are large or the roots are irrational. Completing the square is another method that involves transforming the quadratic equation into a perfect square trinomial. This allows you to isolate x and solve for the roots. While completing the square is a reliable method, it can be more time-consuming than the quadratic formula, especially for equations with complex coefficients. The quadratic formula stands out for its generality. It can be applied to any quadratic equation, regardless of the complexity of the coefficients or the nature of the roots. This makes it a valuable tool when factoring is difficult or impossible, and when completing the square seems cumbersome. However, the quadratic formula can sometimes involve more arithmetic calculations compared to factoring, especially when the numbers are small and easily factorable. The choice of method often depends on the specific equation and your personal preference. For simple equations with easily identifiable factors, factoring might be the most efficient approach. For equations with complex coefficients or irrational roots, the quadratic formula is often the best choice. Completing the square is a valuable method to understand, as it provides insights into the derivation of the quadratic formula itself. In conclusion, while the quadratic formula is a powerful tool, exploring alternative methods like factoring and completing the square can broaden your problem-solving skills and provide you with a more comprehensive understanding of quadratic equations.
Conclusion: Solidifying Your Understanding of the Quadratic Formula
In this comprehensive guide, we have explored the quadratic formula in detail, demonstrating its power and versatility in solving quadratic equations. We began by understanding the significance of the formula and its role in finding the roots of equations in the form ax² + bx + c = 0. We then walked through a step-by-step solution of the equation 12 + 5x(x + 6) = 0, showcasing the practical application of the formula. The process involved transforming the equation into standard form, identifying the coefficients, substituting them into the formula, and simplifying the resulting expression to obtain the exact solutions: x = (-15 + √165) / 5 and x = (-15 - √165) / 5. We emphasized the importance of exact solutions, particularly when dealing with irrational numbers, and discussed the significance of the discriminant in determining the nature of the roots. Furthermore, we compared the quadratic formula with alternative methods like factoring and completing the square, highlighting the strengths and weaknesses of each approach. While factoring can be quicker for simple equations, the quadratic formula provides a universal solution for all quadratic equations, regardless of their complexity. Mastering the quadratic formula is a crucial step in your mathematical journey. It not only equips you with a reliable method for solving quadratic equations but also lays the foundation for more advanced mathematical concepts. By understanding the formula and practicing its application, you can confidently tackle a wide range of problems in various fields, from algebra to calculus and beyond. Remember, the key to success in mathematics lies in consistent practice and a deep understanding of the underlying principles. The quadratic formula is a prime example of such a principle, offering a powerful tool for problem-solving and a gateway to further mathematical exploration.
