Solving X² + 5x - 24 = 0 A Step-by-Step Guide

Introduction to Quadratic Equations

In the realm of mathematics, quadratic equations hold a significant position. These equations, characterized by the presence of a squared variable (x²), pop up in various real-world applications, ranging from physics to engineering. Understanding how to solve them is a fundamental skill. This guide delves into the process of solving the specific quadratic equation x² + 5x - 24 = 0, providing a step-by-step approach and explaining the underlying concepts. The key here is to first understand the components, then apply the appropriate methods to arrive at the solution. Quadratic equations are not just abstract mathematical constructs; they are tools that help us model and solve problems in the world around us. For example, the trajectory of a projectile, the shape of a satellite dish, and even the design of bridges can all be described using quadratic equations. This makes the ability to solve them an essential skill for anyone working in a STEM field.

We will explore the method of using the discriminant to determine the nature of the roots and then apply the quadratic formula to find the actual solutions. This method is not only applicable to this specific equation but also to any quadratic equation of the form ax² + bx + c = 0. By the end of this guide, you will have a solid understanding of how to solve quadratic equations and will be able to apply this knowledge to a wide range of problems. This comprehensive approach will empower you to tackle more complex mathematical challenges and see the beauty and utility of quadratic equations in various fields of study and real-world applications.

Understanding the Given Quadratic Equation

The quadratic equation we aim to solve is x² + 5x - 24 = 0. To begin, let's identify the coefficients. In the standard form of a quadratic equation, which is ax² + bx + c = 0, we can see that:

• a = 1 (the coefficient of x²)
• b = 5 (the coefficient of x)
• c = -24 (the constant term)

These coefficients are crucial for applying the quadratic formula and understanding the properties of the equation's solutions. Recognizing these values is the first step toward solving the equation. The coefficients not only define the shape and position of the parabola represented by the quadratic equation but also dictate the nature of its roots. For instance, the sign and magnitude of the coefficients influence whether the roots are real or complex, distinct or repeated. A thorough understanding of these coefficients is therefore essential for predicting the behavior of the equation and choosing the most appropriate solution method.

Moreover, the constant term 'c' holds significant information about the parabola's intersection with the y-axis. In this case, c = -24 indicates that the parabola intersects the y-axis at the point (0, -24). This visual representation can provide additional insights into the equation's behavior and solutions. By carefully analyzing the coefficients and their relationships, we can gain a deeper understanding of the quadratic equation and its characteristics, paving the way for a more effective solution process. This initial step of identifying and understanding the coefficients is not merely a formality; it is a critical foundation for the rest of the solution process.

Calculating the Discriminant (D)

The discriminant, denoted by D, is a critical component in solving quadratic equations. It helps determine the nature of the roots (solutions) of the equation. The formula for the discriminant is:

D = b² - 4ac

Using the values we identified earlier (a = 1, b = 5, c = -24), we can calculate the discriminant:

D = 5² - 4(1)(-24) D = 25 + 96 D = 121

The discriminant, D = 121, provides valuable information about the roots of the equation. Specifically:

• If D > 0, the equation has two distinct real roots.
• If D = 0, the equation has one real root (a repeated root).
• If D < 0, the equation has two complex roots.

In our case, since D = 121 > 0, we know that the equation x² + 5x - 24 = 0 has two distinct real roots. This knowledge guides us in choosing the appropriate method for finding the solutions. The discriminant acts as a diagnostic tool, providing a crucial insight into the nature of the roots before we even attempt to solve for them. This not only saves time but also ensures that we are using the most efficient approach. For example, if the discriminant were negative, we would know to expect complex solutions and adjust our method accordingly.

The magnitude of the discriminant also gives us an idea of the separation between the roots. A larger positive discriminant suggests that the roots are more widely separated on the number line. In contrast, a smaller positive discriminant indicates that the roots are closer together. The discriminant is, therefore, a powerful tool for understanding the fundamental characteristics of a quadratic equation's solutions, providing a foundation for the subsequent steps in the solving process. Its calculation and interpretation are essential skills for anyone working with quadratic equations.

Applying the Quadratic Formula

Now that we've calculated the discriminant and determined that our equation has two distinct real roots, we can use the quadratic formula to find these roots. The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0 and is given by:

x = (-b ± √D) / 2a

Where D is the discriminant we calculated earlier. Plugging in the values (a = 1, b = 5, D = 121), we get:

x = (-5 ± √121) / 2(1) x = (-5 ± 11) / 2

This gives us two possible solutions:

• x₁ = (-5 + 11) / 2 = 6 / 2 = 3
• x₂ = (-5 - 11) / 2 = -16 / 2 = -8

Therefore, the solutions to the quadratic equation x² + 5x - 24 = 0 are x = 3 and x = -8. The quadratic formula is a cornerstone of algebra, providing a reliable method for solving any quadratic equation regardless of its complexity. Its derivation is rooted in the process of completing the square, a technique that transforms the quadratic equation into a perfect square trinomial. The formula's elegance lies in its ability to encapsulate the solution process in a single, concise expression.

Understanding the quadratic formula is not just about memorizing it; it's about grasping its underlying principles and appreciating its power. The ± sign in the formula highlights the potential for two distinct solutions, reflecting the parabolic nature of quadratic equations. The formula's components, including the coefficients and the discriminant, work together to provide a complete picture of the equation's solutions.

By applying the quadratic formula systematically, we can confidently solve a wide range of quadratic equations, from simple ones like the example in this guide to more complex ones encountered in advanced mathematical contexts. The ability to use the quadratic formula effectively is a valuable asset for anyone pursuing studies or careers in mathematics, science, or engineering.

Verifying the Solutions

To ensure the accuracy of our solutions, it's crucial to verify them by substituting them back into the original equation. Let's verify x = 3:

(3)² + 5(3) - 24 = 9 + 15 - 24 = 0

Now, let's verify x = -8:

(-8)² + 5(-8) - 24 = 64 - 40 - 24 = 0

Both solutions satisfy the original equation, confirming that x = 3 and x = -8 are indeed the correct roots. This verification step is an essential part of the problem-solving process, acting as a safeguard against potential errors. By plugging the solutions back into the original equation, we can be confident that they are accurate and consistent with the equation's properties. Verification not only provides assurance but also deepens our understanding of the relationship between the solutions and the equation itself.

It's a good practice to incorporate verification into any mathematical problem-solving routine, especially when dealing with equations and formulas. This step can help identify mistakes in calculations or algebraic manipulations, preventing the propagation of errors. Moreover, verification can provide valuable insights into the nature of the solutions and their relationship to the problem's context. In the case of quadratic equations, verifying the solutions reinforces the understanding of how the roots satisfy the equation and how they relate to the parabola's x-intercepts.

Therefore, taking the time to verify solutions is not just a matter of prudence; it's an opportunity to enhance comprehension and build confidence in one's mathematical abilities. This habit of verification is a hallmark of a careful and thorough problem solver, contributing to greater accuracy and a deeper appreciation of mathematical concepts.

Conclusion

In this guide, we've systematically solved the quadratic equation x² + 5x - 24 = 0. We began by identifying the coefficients, calculated the discriminant to determine the nature of the roots, applied the quadratic formula to find the solutions, and finally, verified these solutions. This process demonstrates a comprehensive approach to solving quadratic equations, which can be applied to various similar problems. Understanding quadratic equations and mastering the techniques to solve them is a fundamental skill in mathematics with far-reaching applications in science, engineering, and other fields. The ability to solve quadratic equations opens doors to modeling and solving real-world problems, from calculating trajectories in physics to optimizing designs in engineering.

Moreover, the process of solving quadratic equations cultivates critical thinking and problem-solving skills that are valuable in any discipline. The systematic approach, from identifying coefficients to verifying solutions, instills a sense of precision and attention to detail. The use of the discriminant to predict the nature of the roots demonstrates the power of mathematical tools to provide insights into complex problems. The application of the quadratic formula showcases the elegance and efficiency of algebraic methods.

By mastering the techniques presented in this guide, you are not only equipped to solve quadratic equations but also empowered to tackle more advanced mathematical challenges. The journey through this process enhances your understanding of mathematical concepts and fosters a deeper appreciation for the power and beauty of mathematics. This understanding will serve as a solid foundation for future learning and problem-solving endeavors, enabling you to approach complex challenges with confidence and competence.