Solving For Consecutive Odd Integers With A Product Of 143

Introduction: The Puzzle of 143

In the realm of mathematics, integer problems often present us with intriguing puzzles. This article delves into a specific problem involving two positive, consecutive, odd integers with a product of 143. Our mission is to not only complete the equation that represents finding x, the greater integer, but also to unravel the mystery and discover the value of this elusive integer. We will explore the fundamental concepts of consecutive odd integers, algebraic equations, and problem-solving strategies. By the end of this journey, you'll have a comprehensive understanding of how to approach such problems and the satisfaction of finding the solution.

Understanding Consecutive Odd Integers

To begin our quest, we must first understand what consecutive odd integers are. Integers, as you know, are whole numbers, both positive and negative, including zero. Odd integers are those that cannot be divided evenly by 2, leaving a remainder of 1. Examples of odd integers include 1, 3, 5, 7, and so on. Consecutive odd integers are odd integers that follow each other in sequence, with a difference of 2 between them. For example, 3 and 5, 11 and 13, and 25 and 27 are all pairs of consecutive odd integers.

In our problem, we are given two such integers, and we know that their product is 143. This piece of information is crucial, as it allows us to set up an algebraic equation that will lead us to the solution. But before we dive into the equation, let's take a moment to appreciate the beauty of these numbers. Odd integers have a unique charm, often appearing in mathematical patterns and puzzles. Their consecutive nature adds another layer of intrigue, making them perfect candidates for challenging problems.

Formulating the Equation: A Mathematical Representation

Now, let's translate the problem into the language of algebra. We are told that the two integers are consecutive and odd. If we let x represent the greater integer, then the smaller integer must be x - 2. This is because consecutive odd integers differ by 2. The problem also states that the product of these two integers is 143. We can express this mathematically as:

(x - 2) * x = 143

This equation encapsulates the essence of the problem. It represents the relationship between the two consecutive odd integers and their product. By solving this equation, we can find the value of x, the greater integer. However, the equation is not yet in the form we need to complete it. The problem provides the equation as x(x - □) = 143, where we need to fill in the blank. Comparing this to our equation, we can see that the blank represents the difference between the two integers, which we already know is 2. Therefore, the completed equation is:

x(x - 2) = 143

This equation is our key to unlocking the solution. It is a quadratic equation, which means it involves a variable raised to the power of 2. To solve it, we will need to use techniques for solving quadratic equations, such as factoring or using the quadratic formula.

Solving the Equation: Unveiling the Value of x

To solve the equation x(x - 2) = 143, we first need to expand the left side and rearrange the equation into a standard quadratic form, which is _ax_² + bx + c = 0. Expanding the left side, we get:

x_² - 2_x = 143

Now, subtract 143 from both sides to set the equation equal to zero:

x_² - 2_x - 143 = 0

We now have a quadratic equation in standard form. There are several methods to solve quadratic equations, but factoring is often the most efficient if the equation can be factored easily. Factoring involves finding two numbers that multiply to give the constant term (-143) and add up to give the coefficient of the x term (-2). Let's think about the factors of 143. We know that 143 = 11 * 13. Notice that 11 and 13 differ by 2, which is the coefficient of our x term. This suggests that we can factor the quadratic as:

(x - 13)(x + 11) = 0

To verify this, we can expand the factored form: (x - 13)(x + 11) = x_² + 11_x - 13_x_ - 143 = x_² - 2_x - 143. This matches our original quadratic equation, so our factoring is correct.

Now, to find the solutions for x, we set each factor equal to zero:

x - 13 = 0 or x + 11 = 0

Solving these equations, we get:

x = 13 or x = -11

We have two possible solutions for x: 13 and -11. However, the problem specifies that the integers are positive. Therefore, we can discard the solution x = -11. This leaves us with x = 13 as the greater integer.

The Answer: Unveiling the Greater Integer

After our journey through the world of consecutive odd integers and quadratic equations, we have arrived at our destination. We have successfully found the greater integer, x, which is 13. To confirm our answer, let's find the smaller integer and check if their product is indeed 143. The smaller integer is x - 2 = 13 - 2 = 11. The product of 11 and 13 is 11 * 13 = 143, which matches the information given in the problem. Therefore, our solution is correct.

We have not only found the greater integer but also completed the equation to represent finding it. The completed equation is x(x - 2) = 143. This equation, along with our understanding of consecutive odd integers and algebraic techniques, has allowed us to solve the problem and unveil the value of x.

Conclusion: The Power of Mathematical Problem-Solving

In conclusion, we have successfully navigated the problem of finding the greater of two positive, consecutive, odd integers with a product of 143. We began by understanding the concept of consecutive odd integers and then translated the problem into an algebraic equation. We solved the equation using factoring, found two possible solutions, and discarded the negative solution based on the problem's constraints. Finally, we verified our answer and confirmed that the greater integer is indeed 13.

This problem showcases the power of mathematical problem-solving. By combining our knowledge of numbers, algebra, and problem-solving strategies, we can unravel complex puzzles and arrive at elegant solutions. Mathematics is not just about formulas and equations; it's about critical thinking, logical reasoning, and the joy of discovery. As you continue your mathematical journey, remember that every problem is an opportunity to learn, grow, and expand your understanding of the world around you.