Consecutive Odd Integers Product: Solve For The Greater Integer

Hey guys, ever get stumped by a math problem that seems like it's speaking a different language? You know, the kind with words like "consecutive," "odd," and "product" all thrown together? Well, today we're diving deep into a classic problem that'll have you feeling like a math whiz in no time. We're talking about finding two positive, consecutive, odd integers whose product is a neat little number: 143. And the best part? We're going to break down how to complete an equation to find the greater integer, which is represented by '$x$'. Get ready to flex those brain muscles because this is going to be fun!

Understanding Consecutive Odd Integers and Their Product

So, what exactly are we dealing with here? Consecutive odd integers are basically odd numbers that follow each other directly on the number line. Think of numbers like 1, 3, 5, 7, and so on. If you pick any odd number, the next odd number is always two greater than the one before it. For instance, if you have 7, the next consecutive odd integer is 9. If you have -5, the next one is -3. They're always separated by a difference of 2. Now, when we talk about their product, we're just talking about the result you get when you multiply them together. Simple enough, right?

Let's set up our scenario. We're told we have two positive consecutive odd integers. This is important because it narrows down our possibilities. We don't have to worry about negative numbers creeping in. Let's say the smaller of these two integers is '$y$'. Since they are consecutive odd integers, the larger one must be '$y + 2$'. Now, the problem states that their product is 143. So, if we multiply '$y$' by '$y + 2$', we should get 143. This gives us our first equation: $y(y+2) = 143$. This equation is a great starting point, but the problem wants us to work with '$x$', which represents the greater integer. This is a common trick in these types of problems – switching up the variable and what it represents. So, if '$x$' is the greater integer, and the integers are consecutive and odd, what's the smaller integer in terms of '$x$'? It's got to be '$x - 2$', right? Because if '$x$' is the bigger one, and they're separated by 2, the one before it must be 2 less.

Now we can translate the problem into the equation format the question is asking for. The product of the two integers is 143. The greater integer is '$x$', and the smaller integer is '$x - 2$'. So, their product is '$x * (x - 2)$'. Setting this equal to 143, we get the equation: $x(x-2) = 143$. This is exactly what the problem wants us to represent! The equation to find '$x$', the greater integer, is $x(x-2)=143$. Pretty neat how we can transform words into algebraic expressions, huh?

Completing the Equation and Solving for $x$

Alright guys, we've got our equation: $x(x-2) = 143$. Now comes the fun part – solving it to find the value of '$x$', the greater integer. First, let's expand that left side of the equation by distributing the '$x$': $x * x$ gives us $x^2$, and $x * -2$ gives us $-2x$. So, our equation becomes: $x^2 - 2x = 143$.

This looks like a quadratic equation, and it is! To solve a quadratic equation, we usually want to set it equal to zero. So, let's move that 143 over to the left side by subtracting it from both sides: $x^2 - 2x - 143 = 0$. Now we have our standard quadratic form $ax^2 + bx + c = 0$, where $a=1$, $b=-2$, and $c=-143$.

There are a few ways to solve quadratic equations: factoring, using the quadratic formula, or completing the square. For this particular problem, factoring is often the most straightforward if the numbers work out nicely. We need to find two numbers that multiply to '-143' and add up to '-2'. Let's think about the factors of 143. We know 143 isn't divisible by 2 (it's odd), nor 3 (1+4+3=8, not divisible by 3), nor 5. How about 7? 143 / 7 is about 20.something, so nope. Let's try 11. $11 * 10 = 110$, $11 * 3 = 33$. So, $11 * 13 = 143$. Bingo!

Now, we need these two factors to add up to -2. Since we have 11 and 13, and their product is positive 143, we need one of them to be negative for their sum to potentially be negative. If we make the larger number, 13, negative, we get -13. So, let's test our pair: 11 and -13. Their product is $11 * (-13) = -143$. Their sum is $11 + (-13) = -2$. Perfect! These are the numbers we need for factoring.

So, we can rewrite our quadratic equation $x^2 - 2x - 143 = 0$ as $(x + 11)(x - 13) = 0$. To find the possible values of '$x$', we set each factor equal to zero:

1. $x + 11 = 0 ightarrow x = -11$
2. $x - 13 = 0 ightarrow x = 13$

We have two possible solutions for '$x$': -11 and 13. However, the original problem states that we are looking for positive consecutive odd integers. Therefore, we must discard the negative solution. That leaves us with $x = 13$.

So, the greater integer, '$x$', is 13. Let's just double-check this. If the greater integer is 13, then the smaller consecutive odd integer must be $13 - 2 = 11$. Are these two positive? Yes. Are they consecutive odd integers? Yes. Is their product 143? $11 * 13 = 143$. Yes, it all checks out!

Identifying the Greater Integer

We've successfully navigated the algebraic waters and found our solutions for '$x$'. Remember, the equation we were working with, $x(x-2)=143$, was designed to help us find '$x$', which the problem specifically defines as the greater of the two consecutive odd integers. When we solved the quadratic equation $x^2 - 2x - 143 = 0$, we found two potential values for '$x$': -11 and 13.

Now, it's crucial to go back to the problem's constraints. The question explicitly asks for positive consecutive odd integers. This is our key to selecting the correct answer. Out of the two solutions, -11 and 13, only 13 is positive. Therefore, $x = 13$ is our valid solution for the greater integer.

If $x=13$, then the other consecutive odd integer is $x-2$, which is $13-2=11$. Let's verify: are 11 and 13 positive? Yes. Are they consecutive odd integers? Yes, they follow each other on the odd number sequence. Is their product 143? $11 imes 13 = 143$. It all fits perfectly!

So, the greater integer is indeed 13. It's always a good practice, guys, to reread the question after you've found your answer to ensure you've addressed all parts and constraints. We didn't just need to find an integer; we needed to find the greater one, and it had to be positive and part of a consecutive odd pair.

This problem beautifully illustrates how algebra can be used to model and solve real-world (or at least textbook-world!) scenarios. By carefully defining our variables and setting up the correct equation, we can systematically arrive at the solution. The process involved translating words into an algebraic expression, simplifying it into a standard quadratic form, and then employing factoring techniques to find the roots. Finally, we applied the problem's conditions to choose the appropriate root.

Keep practicing these types of problems, and soon you'll be spotting the patterns and solving them with confidence. It's all about breaking it down, understanding the terms, and taking it one step at a time. And remember, if you get two answers, always check back with the original question to see which one makes sense!