Solving The Quadratic Equation 3x² + 24x - 24 = 0 A Step By Step Guide
In this detailed guide, we will explore the process of solving the quadratic equation 3x² + 24x - 24 = 0. Quadratic equations, which are polynomial equations of the second degree, play a crucial role in various fields, including mathematics, physics, engineering, and economics. Mastering the techniques for solving these equations is essential for anyone seeking a strong foundation in these disciplines. This article will provide a step-by-step solution to the given equation, along with a thorough explanation of the methods used. Understanding how to solve quadratic equations is a fundamental skill in algebra, and this guide aims to provide a clear and comprehensive approach to mastering this concept. By breaking down each step and providing detailed explanations, we will ensure that you not only understand the solution but also the underlying principles involved. The goal is to empower you with the knowledge and confidence to tackle similar problems effectively.
Understanding Quadratic Equations
A quadratic equation is defined as an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are also known as the roots or zeros of the equation. These roots represent the x-values at which the parabola represented by the quadratic equation intersects the x-axis. There are several methods for solving quadratic equations, each with its own advantages and applications. The most common methods include factoring, completing the square, and using the quadratic formula. Factoring involves expressing the quadratic expression as a product of two linear factors, which is a straightforward 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, as it transforms the equation into a form where a perfect square trinomial is obtained. The quadratic formula is a powerful tool that provides a direct solution for any quadratic equation, regardless of whether it can be factored or not. It is derived from the method of completing the square and is widely used due to its reliability and efficiency. Each of these methods offers a unique approach to finding the roots of a quadratic equation, and understanding them provides a comprehensive toolkit for solving a variety of problems. The choice of method often depends on the specific characteristics of the equation, such as the values of the coefficients and the ease of factoring.
Simplifying the Equation
The given equation is 3x² + 24x - 24 = 0. Before applying any solving method, it is often beneficial to simplify the equation. This can make the subsequent steps easier and reduce the risk of errors. In this case, we notice that all the coefficients (3, 24, and -24) are divisible by 3. Dividing the entire equation by 3, we get a simpler equivalent equation: x² + 8x - 8 = 0. This simplification does not change the roots of the equation but makes the coefficients smaller and more manageable. Simplifying equations is a crucial step in problem-solving, not just in quadratic equations but in various mathematical contexts. It allows us to work with smaller numbers, reducing the complexity of calculations and making it easier to identify patterns or apply formulas. The simplified equation x² + 8x - 8 = 0 is now in a standard quadratic form, making it easier to apply methods such as completing the square or the quadratic formula. This step highlights the importance of looking for opportunities to simplify equations before proceeding with more complex solution techniques.
Method 1: Completing the Square
Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial, which can then be easily solved. To apply this method to the equation x² + 8x - 8 = 0, we first isolate the constant term on the right side of the equation: x² + 8x = 8. Next, we need to add a value to both sides of the equation that will make the left side a perfect square. This value is determined by taking half of the coefficient of the x term (which is 8), squaring it ( (8/2)² = 16), and adding it to both sides. This gives us x² + 8x + 16 = 8 + 16, which simplifies to x² + 8x + 16 = 24. The left side of the equation is now a perfect square trinomial, which can be factored as (x + 4)². So, the equation becomes (x + 4)² = 24. Taking the square root of both sides, we get x + 4 = ±√24. Simplifying the square root, we have √24 = √(4 * 6) = 2√6. Thus, x + 4 = ±2√6. Finally, we solve for x by subtracting 4 from both sides, giving us x = -4 ± 2√6. This method demonstrates the power of completing the square in transforming a quadratic equation into a solvable form. It is a valuable technique to master, as it provides a systematic approach to solving quadratic equations and is also the foundation for deriving the quadratic formula.
Method 2: Using the Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). For the simplified equation x² + 8x - 8 = 0, we have a = 1, b = 8, and c = -8. Substituting these values into the quadratic formula, we get: x = (-8 ± √(8² - 4 * 1 * -8)) / (2 * 1). Simplifying the expression under the square root: 8² - 4 * 1 * -8 = 64 + 32 = 96. So, the equation becomes: x = (-8 ± √96) / 2. Now, we simplify the square root: √96 = √(16 * 6) = 4√6. Substituting this back into the equation, we have: x = (-8 ± 4√6) / 2. Finally, we divide both terms in the numerator by 2: x = -4 ± 2√6. The quadratic formula provides a direct and efficient way to find the solutions to any quadratic equation, regardless of its factorability. It is an essential tool in algebra and is widely used in various mathematical and scientific applications. Understanding the quadratic formula and its derivation is crucial for a solid foundation in quadratic equations.
The Solutions
Both the method of completing the square and the quadratic formula yield the same solutions for the equation 3x² + 24x - 24 = 0. The solutions are x = -4 + 2√6 and x = -4 - 2√6. These values represent the points where the parabola defined by the quadratic equation intersects the x-axis. In practical terms, these solutions can represent critical points in a system, such as equilibrium points or optimal values. For example, in physics, they might represent the times at which an object reaches a certain position, or in economics, they might represent the points of maximum profit or minimum cost. Understanding the significance of the solutions to a quadratic equation is just as important as knowing how to find them. The solutions provide valuable information about the behavior of the system being modeled by the equation. By analyzing the solutions, we can gain insights into the properties and characteristics of the system, making informed decisions and predictions. This makes the ability to solve quadratic equations a fundamental skill in many fields.
Conclusion
In conclusion, we have successfully solved the quadratic equation 3x² + 24x - 24 = 0 using two different methods: completing the square and the quadratic formula. Both methods lead to the same solutions: x = -4 ± 2√6. This exercise demonstrates the versatility and power of algebraic techniques in solving mathematical problems. Mastering these methods is crucial for anyone studying mathematics, physics, engineering, or related fields. Quadratic equations are ubiquitous in these disciplines, and the ability to solve them efficiently and accurately is essential for success. The steps outlined in this guide provide a clear and comprehensive approach to solving quadratic equations, ensuring that you not only understand the solution but also the underlying principles involved. By practicing these methods and applying them to a variety of problems, you can develop a strong foundation in algebra and gain confidence in your problem-solving abilities. Remember, the key to mastering mathematics is consistent practice and a thorough understanding of the fundamental concepts.
Therefore, the correct answer is C. x = -4 ± 2√6
