Consecutive Odd Integers: Find The Greater Integer

Hey guys! Let's dive into a fun mathematical puzzle today. We're going to explore how to find two positive, consecutive, odd integers that multiply together to give us 143. Sounds intriguing, right? We'll break down the problem step-by-step, making sure everyone understands the process. So, grab your thinking caps, and let's get started!

Setting Up the Equation: A Foundation for Success

To begin, we need to translate the word problem into a mathematical equation. This is a crucial step because it allows us to use the power of algebra to solve the puzzle. Our main keywords here are consecutive odd integers and their product. Remember, consecutive odd integers are odd numbers that follow each other in sequence, like 1 and 3, or 15 and 17. The "product" simply means the result of multiplying these numbers together. Think of these integers as links in a chain, each one connected to the next, creating a numerical relationship that we can unveil. The beauty of setting up an equation is that it transforms a wordy problem into a concise mathematical statement, making it much easier to manipulate and solve. This is where the magic of algebra truly shines, allowing us to represent unknowns with variables and establish relationships between them.

Let's represent the smaller odd integer as x - 2. Since we're looking for consecutive odd integers, the next odd integer will be x. This is because consecutive odd integers differ by 2 (e.g., 3 and 5, 11 and 13). The problem states that the product of these two integers is 143. Therefore, we can write the equation as:

(x - 2) * x = 143

This equation is the heart of our problem. It encapsulates the relationship between the two integers and the given product. Now, our goal is to solve this equation for x, which will give us the greater of the two integers. The process of solving this equation will involve algebraic manipulation, and by understanding each step, we'll not only find the solution but also strengthen our problem-solving skills in mathematics. It's like decoding a secret message, where each step brings us closer to the final answer.

So, the completed equation to represent finding x, the greater integer, is:

x(x - 2) = 143

Cracking the Code: Solving the Quadratic Equation

Now that we have our equation, x(x - 2) = 143, it's time to roll up our sleeves and solve it! First, we need to expand the left side of the equation. This involves distributing the x across the parentheses, which gives us:

x² - 2x = 143

Next, to solve for x, we need to rearrange the equation into a standard quadratic form. A quadratic equation is an equation of the form ax² + bx + c = 0. To get our equation into this form, we subtract 143 from both sides:

x² - 2x - 143 = 0

Now we have a quadratic equation that we can solve. There are a few methods we can use, such as factoring, completing the square, or using the quadratic formula. For this problem, factoring is the most straightforward approach. Factoring involves finding two numbers that multiply to give us -143 and add up to -2. This might sound tricky, but let's break it down.

Think of the factors of 143. We know that 143 is the product of 11 and 13 (11 * 13 = 143). To get a product of -143 and a sum of -2, we can use -13 and 11. This is because (-13) * 11 = -143 and (-13) + 11 = -2. See how we used the factors to find the correct numbers? Therefore, we can factor the quadratic equation as:

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

To solve for x, we set each factor equal to zero:

x - 13 = 0 or x + 11 = 0

Solving these simple equations gives us two possible solutions for x:

x = 13 or x = -11

However, the problem specifies that we are looking for positive integers, so we can discard the solution x = -11. This leaves us with x = 13 as the only valid solution. It's essential to check the solutions against the original problem statement to make sure they fit the given conditions. This step ensures that our answer is not only mathematically correct but also makes sense in the context of the problem.

The Grand Finale: Unveiling the Greater Integer

We've successfully navigated the equation, factored the quadratic, and found the possible values for x. Remember, x represents the greater of the two consecutive odd integers. We found that x = 13 is the valid solution. So, the greater integer is 13. But let's not stop there! It's always a good idea to verify our answer to make sure it's correct. To do this, we can find the smaller odd integer and check if their product is indeed 143.

The smaller odd integer is x - 2, so substituting x = 13, we get:

13 - 2 = 11

Therefore, the two consecutive odd integers are 11 and 13. Now, let's multiply them together:

11 * 13 = 143

Voila! Our solution is correct. The product of the two consecutive odd integers, 11 and 13, is indeed 143. This final step of verification not only confirms our answer but also reinforces our understanding of the problem-solving process. It's like putting the last piece of a puzzle in place, giving us a complete and satisfying picture.

So, the greater integer is 13. We did it!

Wrapping Up: A Triumph of Mathematical Thinking

Guys, we've successfully solved the puzzle! We started with a word problem, translated it into an equation, solved the equation, and verified our answer. This journey showcases the power of mathematical thinking and problem-solving skills. Remember, the key is to break down complex problems into smaller, manageable steps. Each step, from setting up the equation to factoring the quadratic, is a building block in the process. By understanding each step and its purpose, we can tackle even the most challenging problems with confidence.

This problem not only helped us find the solution but also reinforced our understanding of key mathematical concepts such as consecutive odd integers, quadratic equations, and factoring. These concepts are fundamental in algebra and will be valuable in future mathematical endeavors. Always remember to approach problems with a clear and logical mindset, and don't be afraid to explore different approaches. Mathematics is not just about finding the right answer; it's about the journey of discovery and the development of critical thinking skills. So keep practicing, keep exploring, and keep the mathematical spirit alive!

The greater integer is 13. We solved it together, and that's something to be proud of! Keep shining, mathletes!