Equation For Negative Integers 8 Units Apart With A Product Of 308

In the realm of mathematics, problem-solving often involves translating word problems into algebraic equations. This article delves into a specific problem concerning two negative integers, their distance on a number line, and their product. We will explore how to formulate an equation that can determine the smaller of these two integers. This comprehensive exploration aims to provide a clear, step-by-step approach, making it easier to grasp the underlying concepts and apply them to similar problems. Understanding the relationship between the given information and the algebraic representation is crucial for solving such problems efficiently. This article not only focuses on finding the correct equation but also aims to enhance your problem-solving skills in algebra. Let's embark on this mathematical journey, unraveling the complexities and arriving at a solution with clarity and precision.

Problem Statement

The problem presents a scenario involving two negative integers that are 8 units apart on the number line and have a product of 308. The core question is: Which equation could be used to determine x, the smaller negative integer? This type of problem requires a careful translation of the verbal description into algebraic expressions and equations. The key elements to consider are the distance between the integers, their negative nature, and the product they yield. By systematically analyzing these elements, we can construct an equation that accurately represents the problem's conditions. This process involves identifying the variables, understanding the relationships between them, and expressing these relationships in a mathematical form. Let's proceed by dissecting the problem statement and identifying the key components that will guide us toward the correct equation. This methodical approach is essential for tackling mathematical problems effectively and ensuring a thorough understanding of the solution.

Defining Variables

To effectively tackle this problem, we need to define the variables clearly. Let x represent the smaller negative integer. Since the two integers are 8 units apart on the number line, the larger negative integer can be expressed as x + 8. It's crucial to understand why we add 8 rather than subtract. Because we are dealing with negative integers, adding 8 moves us towards zero, thus representing the larger integer. This understanding is fundamental for setting up the correct equation. Misinterpreting this relationship can lead to an incorrect equation and, consequently, a wrong solution. By carefully defining each variable and understanding its relationship to the other, we lay the groundwork for constructing an accurate algebraic representation of the problem. This step is not merely about assigning symbols; it's about capturing the essence of the problem in mathematical terms. With the variables clearly defined, we can now proceed to translate the product condition into an equation.

Formulating the Equation

The problem states that the product of the two integers is 308. We've defined the smaller integer as x and the larger integer as x + 8. Therefore, the product of these two integers can be expressed as x( x + 8). According to the problem, this product equals 308. Hence, we can write the equation as: x( x + 8) = 308. This equation is a direct translation of the problem's condition into algebraic form. It encapsulates the relationship between the two integers and their product. Now, to match the given options, we need to expand and rearrange this equation into a standard quadratic form. This involves distributing x and then moving the constant term to the left side of the equation. The resulting quadratic equation will reveal which of the provided options correctly represents the problem's conditions. This process of algebraic manipulation is a key step in solving the problem and arriving at the correct answer.

Expanding and Rearranging

To transform the equation x(x + 8) = 308 into a standard quadratic form, we first expand the left side. Distributing x across the parentheses gives us x² + 8x = 308. Next, we need to set the equation to zero by subtracting 308 from both sides. This yields the quadratic equation x² + 8x - 308 = 0. This equation is now in the standard form of a quadratic equation, which is ax² + bx + c = 0. Comparing this derived equation with the given options, we can identify the correct one. This step is crucial as it bridges the gap between the initial equation and the answer choices provided. The ability to manipulate equations algebraically is a fundamental skill in mathematics, and this step exemplifies its importance in problem-solving. By carefully expanding and rearranging the equation, we ensure that we are comparing like terms and can confidently select the correct option.

Identifying the Correct Option

After expanding and rearranging the equation, we arrived at x² + 8x - 308 = 0. Now, we compare this equation with the given options:

A. x² - 8x - 308 = 0 B. x² - 8x + 308 = 0 C. x² + 8x + 308 = 0 D. x² + 8x - 308 = 0**

By direct comparison, we can see that option D, x² + 8x - 308 = 0***, exactly matches the equation we derived. Therefore, option D is the correct equation that could be used to determine x, the smaller negative integer. This final step underscores the importance of accurate algebraic manipulation and careful comparison. Selecting the correct option requires a clear understanding of the derived equation and its correspondence to the choices provided. The process of elimination, in this case, further solidifies the correctness of option D. With this, we have successfully identified the equation that represents the problem's conditions.

Conclusion

In conclusion, the problem of determining the equation for two negative integers that are 8 units apart and have a product of 308 was solved by systematically translating the word problem into an algebraic equation. We defined the smaller integer as x, expressed the larger integer as x + 8, and formulated the equation x(x + 8) = 308. By expanding and rearranging this equation, we arrived at the quadratic equation x² + 8x - 308 = 0***, which corresponds to option D. This process highlights the importance of careful variable definition, accurate algebraic manipulation, and meticulous comparison with the given options. The ability to translate word problems into algebraic equations is a fundamental skill in mathematics, and this problem serves as a clear example of its application. By following a structured approach, we can confidently tackle similar problems and arrive at the correct solutions. This exercise not only reinforces algebraic skills but also enhances problem-solving abilities in a broader context.