Finding Consecutive Integers With A Product Of 156
Finding consecutive integers that satisfy a given condition is a common problem in algebra. In this article, we will explore the problem of finding two consecutive integers whose product is 156. This involves setting up an algebraic equation, solving it, and interpreting the solutions in the context of the problem. We will delve into the step-by-step process, offering a detailed explanation that will help clarify the method for solving such problems. This article will also address the significance of considering both positive and negative solutions when dealing with integer problems. Understanding these concepts is crucial for tackling various mathematical challenges, from basic algebra to more advanced topics. Let’s embark on this mathematical journey to unlock the secrets of consecutive integers!
Understanding the Problem
When tackling mathematical problems, it's essential to first understand the problem thoroughly. In this case, we are looking for two consecutive integers. Consecutive integers are numbers that follow each other in order, each differing from the previous number by 1. For example, 1 and 2, 10 and 11, or -5 and -4 are pairs of consecutive integers. The core of the problem lies in finding a pair of these integers such that when they are multiplied together, the result is 156. This requires us to translate the word problem into an algebraic equation, which is a fundamental skill in algebra. Translating word problems into mathematical expressions allows us to use algebraic techniques to find solutions. Understanding the nature of consecutive integers and their properties is crucial for setting up the correct equation. This initial step is the foundation upon which the rest of the solution is built, so careful attention to detail is paramount. By dissecting the problem statement, we can identify the key components and relationships, paving the way for a systematic solution process. Additionally, it is important to consider both positive and negative possibilities, as the product of two negative integers can also be positive. This comprehensive understanding of the problem ensures that we consider all possible solutions and approach the problem with clarity and precision.
Setting up the Equation
To solve this problem, the next step involves setting up an algebraic equation. This is a crucial step in translating the word problem into a mathematical form that we can work with. Let's denote the first integer as 'n'. Since we are looking for consecutive integers, the next integer will be 'n + 1'. The problem states that the product of these two integers is 156. Therefore, we can express this relationship as an equation: n(n + 1) = 156. This equation is a quadratic equation, which means it involves a variable raised to the power of 2. Quadratic equations are commonly encountered in algebra and have specific methods for solving them. Expanding the equation, we get n^2 + n = 156. To solve a quadratic equation, it's standard practice to set it equal to zero. Thus, we subtract 156 from both sides of the equation, resulting in: n^2 + n - 156 = 0. This equation now represents the problem in a standard quadratic form, which we can solve using various methods such as factoring, completing the square, or using the quadratic formula. Setting up the equation correctly is paramount, as it forms the basis for finding the solution. A clear and accurate equation ensures that the subsequent steps will lead to the correct answer. This process of translating a word problem into an algebraic equation demonstrates the power of algebra in solving real-world problems.
Solving the Quadratic Equation
Now that we have the quadratic equation n^2 + n - 156 = 0, the next step is to solve the equation for 'n'. There are several methods to solve quadratic equations, but in this case, factoring is a straightforward approach. Factoring involves breaking down the quadratic expression into two binomials. We are looking for two numbers that multiply to -156 and add up to 1 (the coefficient of the 'n' term). By considering the factors of 156, we find that 13 and -12 satisfy these conditions because 13 * -12 = -156 and 13 + (-12) = 1. Therefore, we can factor the quadratic equation as follows: (n + 13)(n - 12) = 0. To find the solutions for 'n', we set each factor equal to zero: n + 13 = 0 or n - 12 = 0. Solving these equations gives us two possible values for 'n': n = -13 or n = 12. These values represent the first integer in our pair of consecutive integers. Solving quadratic equations is a fundamental skill in algebra, and factoring is one of the most efficient methods when applicable. Understanding the relationship between the factors and the coefficients of the quadratic expression is key to successful factoring. This process demonstrates how algebraic techniques can be used to find specific solutions to mathematical problems.
Finding the Consecutive Integers
After solving the quadratic equation, we found two possible values for 'n': -13 and 12. Now, we need to find the consecutive integers that correspond to each value of 'n'. Recall that the consecutive integer is 'n + 1'. When n = -13, the next consecutive integer is -13 + 1 = -12. So, one pair of consecutive integers is -13 and -12. To verify, we multiply these two integers: -13 * -12 = 156, which matches the given product. Similarly, when n = 12, the next consecutive integer is 12 + 1 = 13. Thus, another pair of consecutive integers is 12 and 13. Multiplying these integers, we get 12 * 13 = 156, which also satisfies the condition. Therefore, we have found two pairs of consecutive integers whose product is 156: 12 and 13, and -13 and -12. Finding the consecutive integers involves using the solutions obtained from the quadratic equation and applying the definition of consecutive integers. This step highlights the importance of interpreting the algebraic solutions in the context of the original problem. By checking the product of each pair, we ensure that our solutions are correct and satisfy the given condition. This process demonstrates the complete cycle of problem-solving, from setting up the equation to finding and verifying the solutions.
Verifying the Solutions
To ensure the accuracy of our solutions, it is crucial to verify the solutions. We found two pairs of consecutive integers: 12 and 13, and -13 and -12. To verify, we will multiply each pair and check if the product is indeed 156. For the first pair, 12 and 13, the product is 12 * 13 = 156, which matches the given condition. For the second pair, -13 and -12, the product is -13 * -12 = 156, which also matches the given condition. Therefore, both pairs of integers satisfy the requirement of having a product of 156. Verifying the solutions is a critical step in the problem-solving process. It helps to catch any errors made during the solution process and ensures that the answers are correct. This step reinforces the importance of accuracy and attention to detail in mathematics. By verifying, we can be confident that our solutions are not only mathematically sound but also relevant to the original problem statement. This practice of verification is applicable to various mathematical problems and is a key component of effective problem-solving strategies.
Conclusion
In conclusion, we have successfully found two pairs of consecutive integers whose product is 156. The pairs are 12 and 13, and -13 and -12. We achieved this by setting up a quadratic equation, solving it by factoring, and then interpreting the solutions in the context of the problem. This problem demonstrates the application of algebraic techniques in solving real-world problems. The process involved translating a word problem into an algebraic equation, solving the equation, and verifying the solutions. Solving problems involving consecutive integers is a common topic in algebra, and understanding the steps involved is essential for mathematical proficiency. By working through this problem, we have reinforced the importance of understanding the problem, setting up the equation correctly, using appropriate solution methods, and verifying the answers. These steps are applicable to a wide range of mathematical problems and highlight the importance of a systematic approach. Furthermore, this exercise underscores the significance of considering both positive and negative solutions when dealing with integers. Mastering these concepts and techniques will empower you to tackle more complex mathematical challenges with confidence and accuracy.
Final Answer: The final answer is (A)
