Finding The Quadratic Equation For Consecutive Integers

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In this article, we will delve into the world of quadratic equations and explore how they can be used to solve problems involving consecutive integers. Specifically, we will tackle the problem of finding the quadratic equation that represents the scenario where the product of two consecutive integers is 342. This is a common type of problem encountered in algebra, and understanding how to set up and solve these equations is crucial for success in mathematics.

Understanding the Problem

Before we dive into the solution, let's first make sure we fully understand the problem. We are given that the product of two consecutive integers is 342. Our goal is to find the quadratic equation that can be used to determine the value of x, where x represents the greater of the two integers. This means we need to translate the word problem into a mathematical equation. This involves identifying the key relationships and representing them using algebraic symbols.

To begin, it's essential to define what consecutive integers are. Consecutive integers are numbers that follow each other in order, each differing from the previous one by 1. For example, 5 and 6 are consecutive integers, as are -3 and -2. In our problem, we have two such integers, and their product is 342. The challenge is to express this relationship in a mathematical form that allows us to solve for the unknown integers.

Setting up the Equation

Now, let's translate the problem into an algebraic equation. We are told that the product of two consecutive integers is 342. Let's define our variables:

  • Let x represent the greater of the two integers.
  • Since the integers are consecutive, the smaller integer will be x - 1.

Now, we can express the given information as an equation. The product of the two integers is (x - 1) multiplied by x, and this product equals 342. So, our equation is:

(x - 1) * x = 342

This equation represents the core relationship described in the problem. To solve for x, we need to manipulate this equation into a standard quadratic form, which is ax² + bx + c = 0. This form allows us to use various methods, such as factoring, completing the square, or the quadratic formula, to find the solutions for x. The next step involves expanding and rearranging the equation to fit this standard form.

Transforming to Quadratic Form

To transform the equation (x - 1) * x = 342 into the standard quadratic form, we need to expand the left side and then rearrange the terms. First, let's expand the left side of the equation:

x * (x - 1) = x² - x

Now, we have:

x² - x = 342

To get the equation into the standard quadratic form (ax² + bx + c = 0), we need to move the constant term (342) to the left side of the equation. We can do this by subtracting 342 from both sides:

x² - x - 342 = 0

This is our quadratic equation in the standard form. Here, a = 1, b = -1, and c = -342. This equation now represents the problem in a format that we can easily work with to find the value of x, the greater of the two consecutive integers. Understanding this transformation is key to solving many algebraic problems involving products and sums of unknown quantities.

Identifying the Correct Option

Now that we have derived the quadratic equation, we can compare it to the options provided in the problem. The equation we found is:

x² - x - 342 = 0

Let's look at the options:

A. x² + 1 = 342 B. x² - 1 = 342 C. x² - x + 342 = 0 D. x² - x - 342 = 0

By comparing our derived equation to the options, we can see that option D, x² - x - 342 = 0, matches exactly. Therefore, option D is the correct quadratic equation that can be used to find x, the greater number.

This step highlights the importance of accurately transforming the word problem into a mathematical equation. Once the equation is in the correct form, identifying the corresponding answer from a set of options becomes a straightforward task. This ability to correctly set up and manipulate equations is a fundamental skill in algebra and problem-solving.

Solving the Quadratic Equation (Optional)

While the problem only asks for the quadratic equation, let's briefly discuss how we could solve this equation to find the value of x. We have the equation:

x² - x - 342 = 0

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring might be a suitable approach. We need to find two numbers that multiply to -342 and add up to -1. These numbers are -19 and 18.

So, we can factor the quadratic equation as follows:

(x - 19)(x + 18) = 0

Now, we can set each factor equal to zero and solve for x:

x - 19 = 0 or x + 18 = 0

x = 19 or x = -18

We have two possible values for x: 19 and -18. Since x represents the greater of the two consecutive integers, we have two pairs of consecutive integers that satisfy the condition:

  • If x = 19, the integers are 18 and 19.
  • If x = -18, the integers are -19 and -18.

This optional step demonstrates how the quadratic equation we found can be used to actually solve for the unknown integers. While not required by the original problem, it provides a complete solution and reinforces the understanding of quadratic equations.

Key Takeaways

  • Translate word problems: The first step in solving any word problem is to translate the given information into mathematical expressions and equations. This involves identifying the unknowns and the relationships between them.
  • Define variables: Clearly define your variables. In this case, we let x represent the greater integer, which allowed us to express the smaller integer as x - 1.
  • Standard quadratic form: Remember the standard form of a quadratic equation (ax² + bx + c = 0). Transforming your equation into this form is crucial for applying various solution methods.
  • Factoring: Factoring is a powerful technique for solving quadratic equations. Look for two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b).
  • Multiple solutions: Quadratic equations can have two solutions. Be sure to consider both possibilities and check if they make sense in the context of the original problem.

By following these steps, you can confidently tackle problems involving consecutive integers and quadratic equations. The ability to translate word problems into mathematical equations is a fundamental skill that will serve you well in algebra and beyond.

Conclusion

In conclusion, we successfully identified the quadratic equation that represents the scenario where the product of two consecutive integers is 342. By carefully translating the word problem into an algebraic equation, expanding and rearranging terms, and comparing our result to the given options, we found that the correct equation is x² - x - 342 = 0. This problem demonstrates the power of quadratic equations in solving real-world problems and reinforces the importance of understanding algebraic concepts.

This step-by-step guide provides a comprehensive approach to solving this type of problem, from understanding the initial conditions to arriving at the final answer. By mastering these techniques, you will be well-prepared to tackle similar problems in algebra and other mathematical contexts. The key is to practice, understand the underlying concepts, and apply them systematically to arrive at the correct solution.