Solving X² - 3x - 28 = 0 Find The Roots

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In this comprehensive guide, we will delve into the process of finding the roots of the quadratic 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 economics. Understanding how to solve them is therefore fundamental. We will explore the different methods available, with a particular focus on the factoring method, to determine which of the given options – a) x = -4, 7, b) x = -7, 4, c) x = -28, 0, or d) x = -24, 4 – correctly identifies the roots of the equation. By the end of this discussion, you will have a clear understanding of how to solve quadratic equations and identify their roots, equipping you with a valuable skill for various mathematical and real-world applications. Remember, mastering the techniques for solving quadratic equations not only enhances your mathematical proficiency but also equips you with problem-solving skills applicable across diverse domains. Let's begin our exploration of solving the quadratic equation x² - 3x - 28 = 0 and unravel the mysteries of its roots. The ability to solve quadratic equations is a cornerstone of mathematical literacy, and with a systematic approach, it becomes a manageable and even enjoyable task.

Understanding Quadratic Equations

To begin, let's first understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, which generally takes the form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The "roots" of a quadratic equation are the values of x that satisfy the equation, meaning that when these values are substituted into the equation, the equation holds true. These roots are also known as the solutions or zeros of the equation. Finding the roots of a quadratic equation is a fundamental problem in algebra, with numerous applications in various fields such as physics, engineering, and economics. The roots represent the points where the parabola defined by the quadratic equation intersects the x-axis. The number of real roots can be two, one, or none, depending on the discriminant (b² - 4ac). Understanding the nature of quadratic equations and their roots is essential for solving problems in various scientific and engineering disciplines. Quadratic equations are not just abstract mathematical concepts; they model real-world phenomena such as projectile motion, optimization problems, and circuit analysis.

Methods for Solving Quadratic Equations

There are several methods for solving quadratic equations, including:

  1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. It is often the simplest method when the quadratic expression can be easily factored.

  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial on one side, which can then be easily solved by taking the square root.

  3. Quadratic Formula: This formula provides a general solution for any quadratic equation and is particularly useful when factoring is difficult or impossible. The formula is given by:

    x = (-b ± √(b² - 4ac)) / 2a

Each method has its strengths and weaknesses, and the choice of method often depends on the specific equation being solved. Factoring is usually the quickest method when it is applicable, but the quadratic formula is the most versatile, as it can be used to solve any quadratic equation. Completing the square is a useful method for deriving the quadratic formula and for understanding the structure of quadratic equations. Understanding these different methods allows us to approach quadratic equations with flexibility and efficiency. Proficiency in these methods is a valuable asset for students and professionals alike. The ability to choose the most appropriate method for a given problem can save time and effort.

Solving x² - 3x - 28 = 0 by Factoring

For the given equation, x² - 3x - 28 = 0, the factoring method is a suitable approach. The goal is to find two numbers that multiply to -28 (the constant term) and add up to -3 (the coefficient of the x term). Let's identify these numbers:

  • We need two numbers with a product of -28 and a sum of -3.
  • Pairs of factors of -28 are: (1, -28), (-1, 28), (2, -14), (-2, 14), (4, -7), and (-4, 7).
  • The pair (4, -7) satisfies the condition since 4 + (-7) = -3.

Now, we can rewrite the quadratic equation using these factors:

x² + 4x - 7x - 28 = 0

Next, we factor by grouping:

x(x + 4) - 7(x + 4) = 0

(x - 7)(x + 4) = 0

To find the roots, we set each factor equal to zero:

x - 7 = 0 or x + 4 = 0

Solving for x gives us:

x = 7 or x = -4

Therefore, the roots of the equation x² - 3x - 28 = 0 are x = -4 and x = 7. This process demonstrates the power of factoring in solving quadratic equations. The ability to recognize factorable quadratic expressions is a key skill in algebra. Factoring not only simplifies the process of finding roots but also provides insights into the structure of the equation. This method is particularly efficient when the coefficients are integers and the roots are rational numbers.

Verifying the Roots

To ensure our solution is correct, we can substitute the roots back into the original equation:

For x = 7:

(7)² - 3(7) - 28 = 49 - 21 - 28 = 0

For x = -4:

(-4)² - 3(-4) - 28 = 16 + 12 - 28 = 0

Both roots satisfy the equation, confirming our solution. This verification step is crucial in mathematics to ensure accuracy. Substituting the solutions back into the original equation provides a reliable check for errors. This practice not only confirms the correctness of the solution but also reinforces the understanding of the relationship between roots and the equation. The verification process is a testament to the rigor of mathematical problem-solving. It adds a layer of confidence to the solution and highlights the importance of precision in mathematical calculations.

Identifying the Correct Option

Based on our calculations, the roots of the equation x² - 3x - 28 = 0 are x = -4 and x = 7. Comparing this with the given options:

a) x = -4, 7 b) x = -7, 4 c) x = -28, 0 d) x = -24, 4

Clearly, option a) x = -4, 7 matches our solution. Therefore, option a) is the correct answer. This final step underscores the importance of careful comparison and attention to detail in problem-solving. Ensuring that the calculated solution aligns perfectly with the given options is essential for arriving at the correct answer. This methodical approach fosters accuracy and minimizes the risk of errors. The ability to identify the correct solution from a set of options is a valuable skill in academic and professional settings.

Conclusion

In conclusion, by applying the factoring method, we have successfully determined that the roots of the quadratic equation x² - 3x - 28 = 0 are x = -4 and x = 7. Therefore, the correct answer is option a) x = -4, 7. This exercise demonstrates the effectiveness of factoring as a method for solving quadratic equations, particularly when the equation can be easily factored. Understanding and applying different methods for solving quadratic equations is a fundamental skill in mathematics, with applications in various fields. Mastering these techniques empowers individuals to tackle complex problems with confidence and precision. The journey of solving quadratic equations not only enhances mathematical proficiency but also cultivates critical thinking and problem-solving skills applicable across diverse domains. Keep practicing and exploring different methods to further strengthen your understanding of quadratic equations and their applications. Remember, mathematics is a journey of discovery, and each solved problem is a step forward in your mathematical growth.