Solving The Quadratic Equation 3x^2 + 24x - 24 = 0 A Step-by-Step Guide

Introduction

In this article, we will dive deep into the process of solving the quadratic equation $3x^2 + 24x - 24 = 0$. Quadratic equations are a fundamental concept in algebra and appear in various fields, including physics, engineering, and economics. Mastering the techniques to solve these equations is crucial for anyone pursuing studies or careers in these areas. We will explore several methods, including factoring, completing the square, and using the quadratic formula, to arrive at the correct solution. Our focus will be on providing a clear, step-by-step explanation, ensuring that readers can follow along and apply these techniques to other quadratic equations they may encounter. Understanding the nuances of quadratic equations not only enhances mathematical proficiency but also develops problem-solving skills that are applicable across various domains. This article aims to break down the complexities of solving quadratic equations, making it accessible and understandable for learners of all levels. We will also highlight common pitfalls and provide tips to avoid errors, ensuring a comprehensive understanding of the topic. By the end of this guide, you should be well-equipped to tackle quadratic equations with confidence and precision. The journey through this equation will not only provide the solution but also illuminate the broader concepts of quadratic functions and their applications. Let's embark on this mathematical exploration together and unravel the solution to this intriguing quadratic equation.

Understanding Quadratic Equations

Before we tackle the given equation, it's essential to understand the basics 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 $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to 0. The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of $x$ that satisfy the equation, making the left-hand side equal to zero. Understanding the nature of these roots is crucial in many applications, from designing parabolic reflectors to modeling projectile motion. The coefficients $a$, $b$, and $c$ play significant roles in determining the shape and position of the parabola represented by the quadratic equation. For instance, the coefficient $a$ dictates whether the parabola opens upwards (if $a > 0$) or downwards (if $a < 0$). The roots can be real or complex, distinct or repeated, depending on the discriminant, which we will discuss later. Quadratic equations can be solved using various methods, each with its advantages and limitations. These methods include factoring, completing the square, and the quadratic formula. Choosing the appropriate method depends on the specific equation and the ease with which it can be applied. A solid grasp of quadratic equations forms the foundation for more advanced mathematical concepts and is indispensable for anyone studying mathematics, physics, or engineering. In the following sections, we will explore these methods in detail and apply them to solve our given equation.

Simplifying the Equation

Our given equation is $3x^2 + 24x - 24 = 0$. The first step in solving any equation is to simplify it as much as possible. In this case, we can observe that all the coefficients are divisible by 3. Dividing the entire equation by 3, we get: $x^2 + 8x - 8 = 0$. This simplification makes the equation easier to work with, especially when using methods like completing the square or the quadratic formula. Simplifying an equation not only reduces the magnitude of the numbers involved but also minimizes the chances of making arithmetic errors. This step is particularly crucial when dealing with larger coefficients or complex equations. By dividing by the greatest common divisor, we ensure that the coefficients are as small as possible, making subsequent calculations more manageable. This simplified form retains all the essential characteristics of the original equation while being more amenable to solution techniques. Moreover, simplification often reveals the underlying structure of the equation, making it easier to identify the most appropriate method for solving it. In this instance, the simplified equation allows us to proceed with confidence, knowing that we have minimized the potential for errors and set the stage for a more straightforward solution process. The art of simplifying equations is a cornerstone of mathematical problem-solving, and this step highlights its importance in the context of quadratic equations. With the simplified equation in hand, we are now ready to explore different methods for finding the roots.

Method 1: Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

x = rac{-b loatPm loatSqrt{b^2 - 4ac}}{2a}

For our simplified equation, $x^2 + 8x - 8 = 0$, we have $a = 1$, $b = 8$, and $c = -8$. Plugging these values into the quadratic formula, we get:

x = rac{-8 loatPm loatSqrt{8^2 - 4(1)(-8)}}{2(1)} x = rac{-8 loatPm loatSqrt{64 + 32}}{2} x = rac{-8 loatPm loatSqrt{96}}{2}

Now, we simplify the square root. Since $96 = 16 imes 6$, we have loatSqrt{96} = loatSqrt{16 imes 6} = 4loatSqrt{6}. Thus,

x = rac{-8 loatPm 4loatSqrt{6}}{2}

Dividing both terms in the numerator by 2, we get:

x = -4 loatPm 2loatSqrt{6}

Thus, the solutions are x = -4 + 2loatSqrt{6} and x = -4 - 2loatSqrt{6}. The quadratic formula provides a systematic approach to solving any quadratic equation, regardless of whether it can be factored easily. It is particularly useful when the roots are irrational or complex. By correctly identifying the coefficients $a$, $b$, and $c$, and carefully substituting them into the formula, we can determine the solutions with precision. The discriminant, $b^2 - 4ac$, within the formula, plays a crucial role in determining the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one repeated real root; and if it is negative, there are two complex roots. In our case, the discriminant is 96, which is positive, indicating two distinct real roots. The quadratic formula's versatility and reliability make it an indispensable tool in solving quadratic equations and forms a cornerstone of algebraic techniques. Its application here demonstrates how it can efficiently handle equations that might be challenging to solve by other methods.

Method 2: Completing the Square

Completing the square is another method for solving quadratic equations. It involves transforming the equation into a perfect square trinomial. Starting with our simplified equation, $x^2 + 8x - 8 = 0$, we first move the constant term to the right side:

$x^2 + 8x = 8$

Next, we need to add a value to both sides of the equation to make the left side a perfect square. The value we need to add is (rac{b}{2})^2, where $b$ is the coefficient of the $x$ term. In this case, $b = 8$, so we add (rac{8}{2})^2 = 4^2 = 16 to both sides:

$x^2 + 8x + 16 = 8 + 16$ $x^2 + 8x + 16 = 24$

Now, the left side is a perfect square trinomial, which can be factored as $(x + 4)^2$:

$(x + 4)^2 = 24$

Taking the square root of both sides, we get:

x + 4 = loatPm loatSqrt{24}

Since $24 = 4 imes 6$, we have loatSqrt{24} = loatSqrt{4 imes 6} = 2loatSqrt{6}. Thus,

x + 4 = loatPm 2loatSqrt{6}

Subtracting 4 from both sides, we get:

x = -4 loatPm 2loatSqrt{6}

Completing the square is a powerful technique that not only solves quadratic equations but also provides insight into the structure of the quadratic function. It transforms the equation into a form that reveals the vertex of the parabola, which is a crucial point in understanding the function's behavior. This method is particularly useful when the quadratic equation cannot be easily factored. By adding the appropriate constant to both sides, we create a perfect square trinomial, which simplifies the process of finding the roots. The steps involved in completing the square—moving the constant term, adding (rac{b}{2})^2, and taking the square root—demonstrate a systematic approach to solving quadratic equations. While it may seem more complex than the quadratic formula at first, completing the square enhances algebraic manipulation skills and provides a deeper understanding of quadratic functions.

Comparing the Solutions

Both the quadratic formula and completing the square methods have yielded the same solutions for the equation $3x^2 + 24x - 24 = 0$. The solutions are:

x = -4 + 2loatSqrt{6} x = -4 - 2loatSqrt{6}

These solutions correspond to option C: x = -4 loatPm 2loatSqrt{6}. The consistency of the solutions obtained through different methods reinforces the accuracy of our calculations and the validity of the techniques used. Each method offers a unique approach to solving quadratic equations, and understanding both enhances problem-solving flexibility. The quadratic formula provides a direct, formulaic approach, while completing the square offers a more conceptual understanding of the equation's structure. The ability to choose the most appropriate method based on the specific problem at hand is a valuable skill in mathematics. Comparing the solutions also allows us to check for errors and ensure that we have not made any mistakes in our calculations. In this case, the agreement between the two methods gives us confidence in the correctness of our answer. Moreover, this comparison highlights the interconnectedness of different mathematical techniques and the importance of having a repertoire of methods to tackle various problems. The solutions not only provide the values of $x$ that satisfy the equation but also represent the x-intercepts of the parabola defined by the quadratic function. These points are critical in analyzing the function's behavior and its applications in real-world scenarios.

Conclusion

In this article, we have thoroughly explored the process of solving the quadratic equation $3x^2 + 24x - 24 = 0$. We began by simplifying the equation and then applied two different methods: the quadratic formula and completing the square. Both methods led us to the same solutions: x = -4 + 2loatSqrt{6} and x = -4 - 2loatSqrt{6}, which corresponds to option C. This exercise demonstrates the power and versatility of algebraic techniques in solving quadratic equations. Understanding these methods not only helps in finding the solutions but also provides a deeper understanding of quadratic functions and their properties. The quadratic formula offers a direct and reliable approach, while completing the square provides insight into the structure of the equation and the vertex of the corresponding parabola. The ability to solve quadratic equations is a fundamental skill in mathematics and is essential for various applications in science, engineering, and other fields. By mastering these techniques, learners can confidently tackle more complex problems and develop a strong foundation in algebra. The journey through this equation has highlighted the importance of simplification, the application of different methods, and the verification of solutions. This comprehensive approach ensures accuracy and builds problem-solving confidence. As we conclude this exploration, remember that practice and understanding the underlying concepts are key to mastering quadratic equations and other mathematical challenges.