Solving Quadratic Equations A Step By Step Guide To X² - 3x - 28 = 0

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Introduction

In this article, we will delve into the process of finding solutions to quadratic equations, with a specific focus on the equation x² - 3x - 28 = 0. Quadratic equations, which are polynomial equations of the second degree, play a crucial role in various fields, including mathematics, physics, engineering, and computer science. Understanding how to solve them is a fundamental skill. This article aims to provide a comprehensive, step-by-step guide on solving the given quadratic equation, ensuring clarity and ease of comprehension for readers of all backgrounds. We will explore different methods for finding solutions, with a particular emphasis on factoring, a technique that simplifies the process significantly for certain types of quadratic equations. Furthermore, we will discuss how these solutions, also known as roots or zeros, represent the points where the parabola defined by the quadratic equation intersects the x-axis. By mastering the techniques presented in this article, readers will gain a solid foundation for tackling more complex mathematical problems and real-world applications involving quadratic equations.

The ability to solve quadratic equations is not just a mathematical exercise; it's a gateway to understanding a wide range of phenomena in the world around us. From calculating projectile trajectories to designing efficient structures, quadratic equations provide the tools necessary to model and solve various problems. This article is designed to demystify the process of finding solutions, making it accessible and engaging for everyone. We encourage readers to actively participate by working through the steps alongside us, reinforcing their understanding and building confidence in their problem-solving abilities. So, let's embark on this journey together and unlock the secrets hidden within the equation x² - 3x - 28 = 0.

Understanding Quadratic Equations

Before diving into the solution, let's first understand what a quadratic equation is. 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 ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis on a graph. The number of real solutions can be zero, one, or two, depending on the nature of the quadratic equation.

In our specific case, the equation x² - 3x - 28 = 0 fits this general form, with a = 1, b = -3, and c = -28. This equation represents a parabola that opens upwards since the coefficient of the x² term (a) is positive. The solutions we seek are the x-coordinates where this parabola crosses the x-axis. Understanding the graphical representation of quadratic equations provides valuable insights into the nature of the solutions. For instance, if the parabola touches the x-axis at only one point, the equation has one real solution (a repeated root). If the parabola does not intersect the x-axis at all, the equation has no real solutions, but it will have two complex solutions. By visualizing the parabola, we can often anticipate the number and nature of the solutions even before we begin the algebraic process of solving the equation.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, including:

  1. Factoring: This method involves expressing the quadratic equation as a product of two binomials. It's often the quickest method when the equation can be factored easily.

  2. Quadratic Formula: This formula provides a direct solution for any quadratic equation, regardless of whether it can be factored easily. The quadratic formula is given by:

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

  3. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial, which can then be easily solved.

For the equation x² - 3x - 28 = 0, we will primarily focus on the factoring method, as it is the most efficient approach for this particular equation. However, it's important to recognize that the quadratic formula is a versatile tool that can be applied to any quadratic equation, even those that are difficult or impossible to factor using simple techniques. The choice of method often depends on the specific characteristics of the equation and the solver's personal preference. Some individuals find factoring more intuitive, while others prefer the straightforward application of the quadratic formula. Ultimately, proficiency in all three methods provides a comprehensive understanding of quadratic equations and their solutions. Completing the square, while less frequently used in isolation, is a fundamental technique that underpins the derivation of the quadratic formula and is valuable for understanding the underlying structure of quadratic expressions.

Solving by Factoring

Factoring involves rewriting the quadratic expression as a product of two binomials. To factor x² - 3x - 28, we need to find two numbers that multiply to -28 (the constant term) and add up to -3 (the coefficient of the x term). Let's analyze the factors of -28:

  • 1 and -28
  • -1 and 28
  • 2 and -14
  • -2 and 14
  • 4 and -7
  • -4 and 7

The pair 4 and -7 satisfy our conditions, as 4 * (-7) = -28 and 4 + (-7) = -3. Therefore, we can rewrite the equation as:

(x+4)(x−7)=0(x + 4)(x - 7) = 0

This step is crucial in the factoring process. We have successfully transformed the quadratic expression into a product of two binomials. Each binomial represents a potential factor that could lead to a solution of the equation. The next step involves applying the zero-product property, which is a fundamental principle in algebra. The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property allows us to break down the problem into simpler equations and solve for the values of x that make each factor equal to zero. By understanding and applying the zero-product property, we can effectively find the solutions to quadratic equations that have been factored.

Now, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero:

x+4=0orx−7=0x + 4 = 0 \quad \text{or} \quad x - 7 = 0

Solving these linear equations, we get:

x=−4orx=7x = -4 \quad \text{or} \quad x = 7

These values of x are the solutions to the quadratic equation. They represent the points where the parabola defined by the equation intersects the x-axis. It's important to verify these solutions by substituting them back into the original equation to ensure they satisfy the equation. This step helps to catch any potential errors made during the factoring or solving process. By verifying the solutions, we can be confident that we have found the correct answers to the quadratic equation. Furthermore, understanding the relationship between the solutions and the graph of the parabola provides a visual confirmation of our results. The solutions correspond to the x-intercepts of the parabola, which can be readily identified on a graph. This connection between algebra and geometry enhances our understanding of quadratic equations and their properties.

Verifying the Solutions

To ensure our solutions are correct, we substitute x = -4 and x = 7 back into the original equation:

For x = -4:

(−4)2−3(−4)−28=16+12−28=0(-4)^2 - 3(-4) - 28 = 16 + 12 - 28 = 0

For x = 7:

(7)2−3(7)−28=49−21−28=0(7)^2 - 3(7) - 28 = 49 - 21 - 28 = 0

Both solutions satisfy the equation, so our solutions are correct.

This verification step is a critical component of the problem-solving process. It serves as a safeguard against potential errors that may have occurred during the algebraic manipulations. By substituting the solutions back into the original equation, we can confirm whether they truly satisfy the equation and are therefore valid. This process not only ensures the accuracy of our answers but also reinforces our understanding of the relationship between the solutions and the equation itself. In the case of quadratic equations, the solutions represent the x-intercepts of the parabola defined by the equation. Verifying these solutions helps us to connect the algebraic representation with the graphical representation, providing a more complete understanding of the problem. Furthermore, the verification step instills confidence in our problem-solving abilities and allows us to move forward with certainty.

Conclusion

Therefore, the solutions to the equation x² - 3x - 28 = 0 are x = -4 and x = 7. Option B, x = -4, 7, is the correct answer. This exercise demonstrates the power of factoring in solving quadratic equations. By breaking down the quadratic expression into a product of binomials, we simplified the problem and found the solutions efficiently. Remember, practice is key to mastering these techniques. The more you practice solving quadratic equations, the more comfortable and confident you will become in applying different methods and finding the solutions. Quadratic equations are a fundamental concept in algebra, and their applications extend far beyond the classroom. They are used in various fields, including physics, engineering, and computer science, to model and solve real-world problems. By mastering the techniques for solving quadratic equations, you are equipping yourself with a valuable tool that will serve you well in your academic and professional pursuits.

In summary, we have explored the process of solving the quadratic equation x² - 3x - 28 = 0 using the factoring method. We began by understanding the general form of a quadratic equation and the significance of its solutions. We then delved into the factoring process, identifying the appropriate factors that would allow us to rewrite the equation as a product of two binomials. By applying the zero-product property, we were able to set each factor equal to zero and solve for the values of x. Finally, we verified our solutions by substituting them back into the original equation, ensuring their accuracy. This step-by-step approach provides a clear and concise method for solving quadratic equations by factoring. We encourage readers to continue practicing these techniques and to explore other methods for solving quadratic equations, such as the quadratic formula and completing the square. A comprehensive understanding of these methods will empower you to tackle a wide range of mathematical problems and to apply these skills in various real-world contexts.