Equation For Two Integers With A Product Of 176

In the realm of mathematics, problem-solving often involves translating word problems into algebraic equations. This article delves into a specific problem concerning two positive integers with a product of 176, where one integer is 5 less than the other. Our primary objective is to identify the equation that accurately represents this scenario and allows us to determine the value of x, the greater integer.

Decoding the Problem: From Words to Equations

Word problems can sometimes appear daunting at first glance, but by carefully dissecting the information provided, we can transform them into manageable mathematical expressions. The key is to identify the unknown quantities and the relationships between them. In this particular problem, we are dealing with two positive integers, let's call them x and y. We are given two crucial pieces of information:

1. The product of the two integers is 176: This translates to the equation x * y* = 176.
2. One integer is 5 less than the other: Assuming x is the greater integer, this can be expressed as y = x - 5.

Our goal is to find an equation that involves only the variable x, the greater integer. To achieve this, we can substitute the expression for y from the second equation into the first equation. This substitution will eliminate y and leave us with an equation in terms of x only. Substituting y = x - 5 into x * y* = 176, we get:

x (x - 5) = 176

This equation, x (x - 5) = 176, is the equation we are looking for. It encapsulates the given information and allows us to solve for x, the greater integer. This equation represents a quadratic equation, which can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Understanding how to translate word problems into algebraic equations is a fundamental skill in mathematics, enabling us to solve a wide range of real-world problems. This process involves identifying the unknowns, defining variables, and expressing the relationships between the variables in mathematical terms. By carefully analyzing the problem statement and breaking it down into smaller parts, we can effectively bridge the gap between words and equations.

Evaluating the Answer Choices: Finding the Perfect Fit

Now, let's examine the answer choices provided and determine which one matches the equation we derived:

A. x^2 + 5 = 176 B. x (x - 5) = 176 C. x^2 - 5 = 176 D. x (x + 5) = 176

By comparing these options with our derived equation, x (x - 5) = 176, we can clearly see that option B is the correct answer. The other options do not accurately represent the relationship between the two integers as described in the problem statement. Option A, x^2 + 5 = 176, incorrectly adds 5 to the square of x. Option C, x^2 - 5 = 176, subtracts 5 from the square of x, which is not consistent with the problem's conditions. Option D, x (x + 5) = 176, adds 5 to x before multiplying, which also does not align with the given information. Therefore, only option B, x (x - 5) = 176, accurately captures the relationship between the two integers, where one integer is 5 less than the other and their product is 176. Choosing the correct equation is a critical step in solving mathematical problems. It requires a thorough understanding of the problem statement and the ability to translate the given information into algebraic expressions. By carefully evaluating each answer choice and comparing it with the derived equation, we can confidently identify the correct answer and proceed with solving for the unknown variable.

Delving Deeper: Solving the Equation and Finding the Integers

While the problem asks for the equation, let's go a step further and solve the equation to find the actual values of the integers. Expanding the equation x (x - 5) = 176, we get:

x^2 - 5x = 176

To solve this quadratic equation, we need to rearrange it into the standard form ax^2 + bx + c = 0:

x^2 - 5x - 176 = 0

Now, we can solve this quadratic equation by factoring. We are looking for two numbers that multiply to -176 and add up to -5. These numbers are -16 and 11. Therefore, we can factor the quadratic equation as:

(x - 16) (x + 11) = 0

This gives us two possible solutions for x:

x - 16 = 0 => x = 16 x + 11 = 0 => x = -11

Since the problem specifies that the integers are positive, we discard the negative solution x = -11. Therefore, the greater integer, x, is 16. To find the other integer, y, we can use the equation y = x - 5:

y = 16 - 5 = 11

So, the two positive integers are 16 and 11. We can verify that their product is indeed 176 (16 * 11 = 176) and that one integer is 5 less than the other (11 = 16 - 5). Solving the equation not only provides the value of the unknown variable but also reinforces the understanding of the problem and the relationships between the given quantities. This process involves applying algebraic techniques, such as expanding, rearranging, and factoring, to arrive at the solution. By verifying the solution, we can ensure its accuracy and gain confidence in our problem-solving abilities.

Key Concepts and Strategies: Mastering Word Problems

This problem highlights several key concepts and strategies that are essential for mastering word problems in mathematics:

• Translating Words into Equations: The ability to convert word problems into mathematical equations is fundamental. This involves identifying the unknowns, assigning variables, and expressing the relationships between them using mathematical symbols and operations.
• Identifying Key Information: Carefully read the problem statement and extract the crucial information. This includes the given quantities, the relationships between them, and the question being asked.
• Defining Variables: Clearly define the variables you will use to represent the unknown quantities. This will help you keep track of the information and avoid confusion.
• Formulating Equations: Use the given information to write equations that relate the variables. This may involve using algebraic formulas, geometric principles, or other mathematical concepts.
• Solving Equations: Apply appropriate algebraic techniques to solve the equations and find the values of the unknown variables. This may involve simplifying, rearranging, factoring, or using formulas.
• Verifying Solutions: After finding a solution, verify that it satisfies the conditions of the problem. This will help you catch any errors and ensure the accuracy of your answer.

By mastering these concepts and strategies, you can approach word problems with confidence and solve them effectively. Practice is key to developing these skills, so make sure to work through a variety of problems to solidify your understanding. The more you practice, the better you will become at recognizing patterns, identifying key information, and translating words into equations. This will not only improve your problem-solving abilities in mathematics but also enhance your critical thinking and analytical skills in other areas of life.

Conclusion: The Power of Algebraic Representation

In conclusion, the equation x (x - 5) = 176 accurately represents the given problem, where two positive integers have a product of 176 and one integer is 5 less than the other. By solving this equation, we found that the greater integer is 16 and the other integer is 11. This problem demonstrates the power of algebraic representation in solving word problems. By translating the problem into an equation, we can use algebraic techniques to find the solution. The ability to translate word problems into mathematical expressions is a valuable skill that is applicable in various fields, including science, engineering, finance, and everyday life. This skill involves understanding the problem's context, identifying the unknowns, and expressing the relationships between them in a concise and accurate manner. Furthermore, the process of solving equations reinforces our understanding of mathematical concepts and enhances our problem-solving abilities. By approaching word problems with a systematic and logical approach, we can effectively break them down into smaller, more manageable parts and arrive at the correct solution. This involves careful reading, identifying key information, defining variables, formulating equations, and verifying solutions. With practice and dedication, anyone can master the art of solving word problems and unlock the power of mathematics.