Solving 4x² + 12x - 72 = 0 A Step-by-Step Guide Using The Quadratic Formula
Introduction to Quadratic Equations
In the realm of mathematics, quadratic equations hold a significant place. These equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, appear in various scientific and engineering applications. Understanding how to solve these equations is crucial for anyone delving into algebra and beyond. One of the most versatile methods for finding the solutions (also known as roots) of a quadratic equation is the quadratic formula. This article will provide a comprehensive guide on using the quadratic formula to solve a specific equation, 4x² + 12x - 72 = 0. We will break down each step, ensuring a clear and thorough understanding of the process. The quadratic formula is a powerful tool because it can solve any quadratic equation, regardless of whether it can be factored easily or not. This makes it an indispensable part of any mathematician's toolkit. This article is tailored for students, educators, and anyone looking to refresh their algebra skills, providing a step-by-step approach that makes solving quadratic equations accessible and straightforward.
The importance of understanding quadratic equations extends beyond the classroom. They are used in physics to model projectile motion, in engineering to design structures, and in finance to calculate growth rates. Mastering the techniques to solve these equations opens up a world of possibilities in various fields. The quadratic formula, in particular, is not just a formula to memorize; it's a concept to understand. By grasping the underlying principles, you can apply it confidently to a wide range of problems. In the following sections, we will explore the quadratic formula in detail, apply it to the equation 4x² + 12x - 72 = 0, and discuss the different types of solutions that quadratic equations can have. By the end of this article, you will have a solid foundation in solving quadratic equations and be well-equipped to tackle more complex algebraic challenges.
The journey of solving quadratic equations can sometimes seem daunting, but with a structured approach and a clear understanding of the quadratic formula, it becomes a manageable and even enjoyable process. This article aims to demystify the process and empower you to confidently solve quadratic equations. We will focus not only on the mechanics of using the formula but also on the underlying concepts that make it work. This holistic approach ensures that you are not just memorizing steps but truly understanding the mathematics involved. So, let's embark on this mathematical journey together and unlock the power of the quadratic formula.
Understanding the General Quadratic Formula
The general quadratic formula is a cornerstone in algebra, providing a universal method to solve any quadratic equation. The formula is derived from the standard form of a quadratic equation, which is ax² + bx + c = 0, where a, b, and c are coefficients representing constants, and x is the variable we aim to solve for. Crucially, a cannot be zero, otherwise, the equation would reduce to a linear equation. The quadratic formula itself is expressed as follows:
x = (-b ± √(b² - 4ac)) / (2a)
This formula provides two possible solutions for x, represented by the ± symbol. The solutions, often called roots or zeros, are the values of x that satisfy the original equation. The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant plays a vital 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). And if it is negative, the equation has two complex solutions. Understanding the discriminant is key to interpreting the solutions obtained from the quadratic formula.
The power of the quadratic formula lies in its ability to handle any quadratic equation, regardless of the complexity of the coefficients. While some quadratic equations can be solved by factoring or completing the square, the quadratic formula provides a reliable and straightforward method that works in all cases. This makes it an essential tool for students, mathematicians, and anyone working with algebraic equations. The formula is not just a set of symbols; it's a representation of a fundamental mathematical relationship. By understanding its derivation and application, you gain a deeper appreciation for the structure of quadratic equations and the nature of their solutions. In the following sections, we will apply the quadratic formula to a specific example, demonstrating how to use it step-by-step to find the solutions of a given equation.
Furthermore, the quadratic formula is more than just a calculation tool; it is a gateway to understanding the properties of quadratic functions and their graphs. The solutions obtained from the formula correspond to the x-intercepts of the parabola represented by the quadratic equation. The vertex of the parabola, which represents the maximum or minimum point of the function, can also be determined using the coefficients of the quadratic equation. Thus, mastering the quadratic formula not only equips you to solve equations but also provides valuable insights into the broader landscape of quadratic functions and their applications in various fields.
Applying the Quadratic Formula to 4x² + 12x - 72 = 0
Now, let's apply the quadratic formula to the specific equation 4x² + 12x - 72 = 0. The first step is to identify the coefficients a, b, and c from the equation. In this case, a = 4, b = 12, and c = -72. It's crucial to correctly identify these coefficients, including their signs, as they will be directly substituted into the quadratic formula. Once the coefficients are identified, we can substitute them into the formula:
x = (-b ± √(b² - 4ac)) / (2a)
Substituting a = 4, b = 12, and c = -72, we get:
x = (-12 ± √(12² - 4 * 4 * -72)) / (2 * 4)
The next step is to simplify the expression. First, we calculate the discriminant, which is the expression inside the square root:
Discriminant = 12² - 4 * 4 * -72 = 144 + 1152 = 1296
Since the discriminant is positive (1296), we know that the equation has two distinct real solutions. Now, we can continue simplifying the quadratic formula:
x = (-12 ± √1296) / 8
The square root of 1296 is 36, so we have:
x = (-12 ± 36) / 8
This gives us two possible solutions:
x₁ = (-12 + 36) / 8 = 24 / 8 = 3
x₂ = (-12 - 36) / 8 = -48 / 8 = -6
Therefore, the solutions to the quadratic equation 4x² + 12x - 72 = 0 are x = 3 and x = -6. These values are the roots of the equation, meaning they are the values of x that make the equation true. We can verify these solutions by substituting them back into the original equation. If we substitute x = 3, we get 4*(3)² + 12*(3) - 72 = 36 + 36 - 72 = 0, which confirms that x = 3 is a solution. Similarly, substituting x = -6, we get 4*(-6)² + 12*(-6) - 72 = 144 - 72 - 72 = 0, confirming that x = -6 is also a solution.
This step-by-step application of the quadratic formula demonstrates its power and versatility in solving quadratic equations. By carefully identifying the coefficients, substituting them into the formula, and simplifying the expression, we can confidently find the solutions to any quadratic equation. The example above highlights the importance of paying attention to signs and performing calculations accurately to arrive at the correct solutions.
Verifying the Solutions
After obtaining the solutions using the quadratic formula, it is crucial to verify them to ensure accuracy. Verification involves substituting the solutions back into the original equation to check if they satisfy it. This step helps catch any errors made during the calculation process and confirms the correctness of the solutions. For the equation 4x² + 12x - 72 = 0, we found the solutions to be x = 3 and x = -6. Let's verify these solutions.
First, we'll substitute x = 3 into the original equation:
4*(3)² + 12*(3) - 72 = 4*(9) + 36 - 72 = 36 + 36 - 72 = 72 - 72 = 0
Since the equation holds true when x = 3, we can confirm that x = 3 is indeed a solution.
Next, we'll substitute x = -6 into the original equation:
4*(-6)² + 12*(-6) - 72 = 4*(36) - 72 - 72 = 144 - 72 - 72 = 144 - 144 = 0
Again, the equation holds true when x = -6, confirming that x = -6 is also a solution.
By verifying the solutions, we have gained confidence in their correctness. This process is a vital part of solving any mathematical problem, especially quadratic equations. It not only ensures accuracy but also reinforces the understanding of the relationship between the solutions and the equation itself. In practical applications, verifying solutions is particularly important, as errors in calculations can have significant consequences. Therefore, it is a good practice to always verify your solutions, regardless of the method used to obtain them.
Moreover, the act of verification provides an opportunity to deepen your understanding of the equation and its solutions. It allows you to see firsthand how the solutions relate to the original problem and why they work. This can be particularly helpful for students who are learning about quadratic equations for the first time. By taking the time to verify your solutions, you are not just checking for errors; you are also reinforcing your understanding of the underlying mathematical principles.
Alternative Methods for Solving Quadratic Equations
While the quadratic formula provides a universal method for solving quadratic equations, there are alternative approaches that can be more efficient in certain cases. Two common methods are factoring and completing the square. Factoring involves expressing the quadratic expression as a product of two linear factors. This method is particularly useful when the coefficients of the quadratic equation are integers and the roots are rational numbers. Completing the square, on the other hand, involves manipulating the quadratic equation to create a perfect square trinomial, which can then be easily solved by taking the square root of both sides. This method is especially useful when the quadratic equation cannot be easily factored.
For the equation 4x² + 12x - 72 = 0, we can also consider factoring as an alternative method. First, we can simplify the equation by dividing all terms by the greatest common divisor, which is 4:
x² + 3x - 18 = 0
Now, we look for two numbers that multiply to -18 and add to 3. These numbers are 6 and -3. Therefore, we can factor the quadratic expression as:
(x + 6)(x - 3) = 0
Setting each factor equal to zero, we get:
x + 6 = 0 or x - 3 = 0
Solving for x, we find the solutions x = -6 and x = 3, which are the same solutions we obtained using the quadratic formula. This demonstrates that factoring can be a faster method for solving certain quadratic equations.
Completing the square is another valuable technique. Although it might seem more complex at first, it provides a deep understanding of the structure of quadratic equations and is essential for deriving the quadratic formula itself. It involves rewriting the quadratic equation in the form (x + p)² = q, where p and q are constants. This form allows us to easily solve for x by taking the square root of both sides. While factoring is often quicker when applicable, completing the square and the quadratic formula are reliable methods that work for all quadratic equations.
Understanding these alternative methods not only broadens your problem-solving toolkit but also enhances your appreciation for the versatility of algebraic techniques. Each method offers a unique perspective on quadratic equations, and choosing the most appropriate method depends on the specific characteristics of the equation at hand.
Conclusion
In conclusion, solving the quadratic equation 4x² + 12x - 72 = 0 using the quadratic formula provides a comprehensive illustration of this fundamental algebraic technique. We have walked through the process step-by-step, from identifying the coefficients a, b, and c, to substituting them into the formula, simplifying the expression, and obtaining the solutions x = 3 and x = -6. Furthermore, we emphasized the crucial step of verifying the solutions by substituting them back into the original equation, ensuring their accuracy. This meticulous approach not only guarantees the correctness of the solutions but also reinforces the understanding of the underlying principles of quadratic equations.
The quadratic formula stands as a powerful tool in the mathematician's arsenal, capable of solving any quadratic equation, regardless of its complexity. While alternative methods like factoring and completing the square can be more efficient in specific cases, the quadratic formula provides a reliable and universally applicable method. Its importance extends beyond the classroom, finding applications in various fields such as physics, engineering, and finance.
Mastering the quadratic formula is not merely about memorizing a formula; it's about developing a deeper understanding of quadratic equations and their solutions. By grasping the underlying concepts and practicing the application of the formula, one can confidently tackle a wide range of algebraic problems. This article aimed to demystify the process, providing a clear and structured approach to solving quadratic equations. The ability to solve quadratic equations is a valuable skill that empowers individuals to analyze and model real-world phenomena, making it an essential component of mathematical literacy.
Ultimately, the journey of solving quadratic equations is a journey of mathematical discovery. It's a process of exploration, where we apply our knowledge, hone our skills, and gain a deeper appreciation for the beauty and power of mathematics. The quadratic formula serves as a guide, illuminating the path to understanding and empowering us to solve the challenges that lie ahead. By embracing this journey, we unlock our potential and expand our horizons in the world of mathematics and beyond. The solutions we find are not just numbers; they are keys to unlocking deeper insights and fostering a lifelong love for learning.
