Finding Consecutive Odd Integers With A Product Of 143

Hey guys! Ever stumbled upon a math problem that seems like a puzzle? Well, let's dive into one today that involves positive, consecutive, odd integers. We're given that their product is 143, and our mission is to find the greater of these two integers. Sounds like a fun challenge, right? So, let's break it down and solve this mystery together!

Setting Up the Equation

To kick things off, we need to represent these integers algebraically. Since we're dealing with consecutive odd integers, we can define them using a variable. Let's say the smaller odd integer is x - 2, and since they are consecutive odd numbers, the next odd integer would be x. The problem tells us that their product is 143. So, we can write this relationship as an equation:

(x)(x - 2) = 143

This equation is the key to unlocking our problem. It represents the mathematical relationship between the two consecutive odd integers and their product. Now, let's dive deeper into why we chose this representation and how it helps us solve the problem.

Understanding Consecutive Odd Integers

Before we jump into solving the equation, let's take a moment to really understand what consecutive odd integers are. Think of odd numbers like 1, 3, 5, 7, and so on. They're the numbers that can't be divided evenly by 2. Consecutive odd integers are simply odd numbers that follow each other in sequence. For example, 3 and 5, or 11 and 13. The crucial thing to notice is that there's always a difference of 2 between them. This is why we represent them as x and x - 2 in our equation.

By using this representation, we ensure that we're always dealing with two odd integers that are consecutive. If x is an odd integer, then x - 2 will also be an odd integer, and they'll be right next to each other in the sequence of odd numbers. This understanding is vital for setting up the equation correctly and solving for our unknown integers.

Expanding the Equation

Now that we have our equation, (x)(x - 2) = 143, let's expand it to get a more familiar form. We'll use the distributive property to multiply x by both terms inside the parentheses:

x * x - x * 2 = 143

This simplifies to:

x² - 2x = 143

Now we have a quadratic equation, which is a type of equation we can solve using various methods. The next step is to rearrange the equation so that it's equal to zero. This is a standard form for quadratic equations, and it allows us to use techniques like factoring or the quadratic formula to find the solutions. So, let's move the 143 to the left side of the equation.

Setting the Stage for Solving: Transforming to Standard Quadratic Form

To solve for x, we need to set our quadratic equation to zero. This means subtracting 143 from both sides of the equation:

x² - 2x - 143 = 0

Now we have a quadratic equation in standard form, which is ax² + bx + c = 0. In our case, a is 1, b is -2, and c is -143. This form is super useful because it allows us to use several methods to find the values of x that satisfy the equation. We can try factoring, which involves finding two numbers that multiply to c and add up to b. Or, we can use the quadratic formula, a trusty tool that works for any quadratic equation. Let's explore the factoring method first.

Unraveling the Mystery: Solving the Quadratic Equation

With our equation in the form x² - 2x - 143 = 0, we can now solve for x. Let's use the factoring method first. Factoring involves finding two numbers that, when multiplied, give us -143 (the constant term) and, when added, give us -2 (the coefficient of the x term). Think of it like this: we're trying to break down the quadratic expression into two binomials that, when multiplied, give us the original expression.

Factoring the Quadratic

So, what two numbers multiply to -143 and add up to -2? After a bit of thinking (or perhaps listing out factors of 143), we'll find that -13 and 11 fit the bill perfectly. -13 multiplied by 11 is -143, and -13 plus 11 is -2. Great! This means we can rewrite our quadratic equation in factored form:

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

Now we have two factors that multiply to zero. This is a crucial step because of the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible solutions for x.

Finding the Possible Solutions

Using the zero-product property, we can set each factor equal to zero and solve for x:

• x - 13 = 0 => x = 13
• x + 11 = 0 => x = -11

So, we have two possible values for x: 13 and -11. But remember, the problem stated that we're looking for positive integers. This means we can discard the negative solution, -11. So, x = 13 is the value we're interested in.

Identifying the Greater Integer and Completing the Equation

We've found that x = 13, but what does this mean in the context of our problem? Remember, x represents the greater of the two consecutive odd integers. So, 13 is the greater integer we were looking for! Woohoo! Now, let's backtrack a bit to the original equation we were given:

x(x - □) = 143

We need to figure out what number goes in the box. We know that the two consecutive odd integers are x and x - 2. Since x is the greater integer, x - 2 is the smaller one. Therefore, the number that goes in the box is 2. This makes our equation:

x(x - 2) = 143

Putting It All Together

Let's recap what we've done. We started with the problem of finding two positive, consecutive, odd integers whose product is 143. We represented these integers algebraically as x and x - 2, set up the equation x(x - 2) = 143, and solved for x. We found two possible solutions, 13 and -11, but discarded the negative solution because we were looking for positive integers. We then identified 13 as the greater integer and completed the equation. So, to summarize:

• The equation is: x(x - 2) = 143
• The greater integer is: 13

The Answer: Greater Integer Unveiled

So, there you have it! The greater integer is 13. We successfully navigated through the problem, setting up the equation, solving it, and interpreting the results. Math problems like these might seem daunting at first, but by breaking them down into smaller steps and understanding the underlying concepts, we can conquer them! Keep practicing, keep exploring, and most importantly, keep having fun with math!

Final Answer: The greater integer is 13.