Equation For Negative Integers 8 Units Apart Product 308

Jul 24, 2025 by ADMIN 57 views

Decoding the Math Puzzle Finding the Equation for Two Negative Integers

Hey guys! Ever stumbled upon a math problem that feels like a riddle wrapped in an enigma? Well, let's untangle one such puzzle today. We're diving into a problem involving negative integers, their distance on the number line, and their product. Buckle up, it's gonna be a fun ride!

The Challenge: Two Negative Integers and a Mysterious Equation

So, here's the deal: we have two negative integers chilling on the number line, 8 units apart. Their product? A solid 308. The mission, should you choose to accept it, is to figure out which equation can help us pinpoint the smaller of these negative integers, which we're calling x. We've got four potential equations lined up as suspects:

A. $x^2 + 8x - 308 = 0$
B. $x^2 - 8x + 308 = 0$
C. $x^2 + 8x + 308 = 0$
D. $x^2 - 8x - 308 = 0$

Now, before we dive headfirst into solving, let's break down the problem and map out our strategy. Understanding the problem is half the battle, right?

Cracking the Code: Understanding the Problem and Strategy

The heart of this problem lies in translating the word puzzle into mathematical language. We're dealing with two negative integers, and here's what we know:

Distance: They're 8 units apart on the number line. This tells us there's a difference of 8 between them.
Product: When we multiply these two integers, we get 308.

Our mission is to craft an equation that captures this information. We'll start by representing the two integers algebraically. Since x is the smaller negative integer, the larger one will be x + 8 (remember, we're moving right on the number line, so the number is greater, but since they are both negative numbers, the one closest to zero is larger). Now, let's translate the product information into an equation.

To create an equation, consider the core mathematical principles at play. Here, the key is the relationship between the two integers. If x represents the smaller integer, then the larger integer, being 8 units apart, can be represented as x + 8. The product of these integers is given as 308, meaning x(x + 8) = 308*. This equation forms the foundation for finding the correct quadratic equation among the options. This step involves translating the problem's conditions into an algebraic form, ensuring accurate representation of the relationships between the unknown integers. Accurate interpretation and algebraic translation are crucial in setting up the equation correctly. This foundation will guide us in solving for x, the smaller negative integer, by accurately reflecting the problem's conditions in an algebraic form. This ensures that when we solve the quadratic equation, we are indeed finding the values that satisfy the original problem statement about the two negative integers and their relationship.

Building the Equation: From Words to Algebra

We know the product of our two integers, x and x + 8, is 308. So, we can write this as:

x(x + 8) = 308

Now, let's expand this equation and see if we can match it to one of our options. Expanding gives us:

$x^2 + 8x = 308$

To get this into the standard quadratic equation form (which looks like $ax^2 + bx + c = 0$ ), we need to subtract 308 from both sides:

$x^2 + 8x - 308 = 0$

And bingo! This matches option A. But hold on, we're not done yet. We need to understand why this is the correct equation and what the other options are trying to trick us with.

To deepen our understanding, let's analyze the transformation from the initial problem statement to the final quadratic equation. The initial equation, x(x + 8) = 308, directly represents the product of the two integers being equal to 308. Expanding this equation and rearranging it into the standard quadratic form, x² + 8x - 308 = 0, allows us to apply standard methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Each term in the quadratic equation plays a crucial role. The x² term represents the square of the smaller integer, the 8x term comes from multiplying x by the difference between the integers, and the constant term -308 is the negative of their product. Understanding these components helps in appreciating how the equation encapsulates the problem's conditions. This transformation from a simple product equation to a quadratic equation highlights the power of algebraic manipulation in solving mathematical problems. It allows us to bring the problem into a form where standard techniques can be applied, making the solution process more accessible and systematic. This step is a key demonstration of how algebraic skills can be used to tackle real-world problems that can be modeled mathematically. It reinforces the idea that mathematical equations are not just abstract symbols but powerful tools for representing and solving complex situations.

Why Option A is the Hero: Unpacking the Correct Equation

Option A, $x^2 + 8x - 308 = 0$ , is the equation that perfectly captures the problem's conditions. Let's break it down:

$x^2$ : This term comes from multiplying x by itself, part of the product of our two integers.
+8x: This represents 8 times x, which arises from multiplying x by the difference between the two integers (which is 8).
-308: This is the key! The negative sign is crucial because it reflects the fact that when we moved the 308 to the left side of the equation, we subtracted it. It represents the negative of the product of the two integers, which is essential for setting up the equation to equal zero.

This equation sets the stage for solving for x, the smaller negative integer. By finding the roots of this quadratic equation, we'll uncover the value of x that satisfies the conditions of our problem.

To further illustrate why Option A is the hero, let's consider what solving this equation would entail. The solutions to the quadratic equation x² + 8x - 308 = 0 will be the values of x that make the equation true. These values will represent the smaller of the two negative integers we are looking for. We can use various methods to solve the equation, such as factoring, completing the square, or applying the quadratic formula. For example, if we were to factor the quadratic equation, we would look for two numbers that multiply to -308 and add up to 8. These numbers would help us rewrite the equation in a factored form, from which we can easily find the roots. The roots of the equation will be the possible values for x, and since we are looking for a negative integer, we would choose the negative root. This negative root will be the smaller negative integer that satisfies the conditions of the problem. This process demonstrates how the quadratic equation is a powerful tool for solving mathematical problems, allowing us to find the values that meet the given criteria. The fact that Option A leads us to this process confirms its role as the hero in this mathematical challenge.

Decoding the Villains: Why the Other Options Fall Short

Now, let's put on our detective hats and figure out why options B, C, and D are the wrongdoers in this mathematical mystery.

Option B: $x^2 - 8x + 308 = 0$ – This equation has a -8x term and a +308. The -8x would suggest a difference of -8 between the integers, which isn't what the problem states. The +308 is also a red flag because we need a negative constant term to represent the product correctly.
Option C: $x^2 + 8x + 308 = 0$ – The +308 here is the culprit. Just like in option B, we need a negative constant term to reflect the subtraction we did when setting the equation to zero.
Option D: $x^2 - 8x - 308 = 0$ – This one's tricky! It has the correct negative constant term, but the -8x term throws us off again. It implies a difference of -8 between the integers, which contradicts the problem statement.

By carefully analyzing each term in the equations, we can see why only Option A accurately represents the relationships between our two negative integers.

Let's delve deeper into why options B, C, and D are incorrect, focusing on the subtle mathematical errors they represent. Option B, with the equation x² - 8x + 308 = 0, introduces a significant error in the sign of the 8x term. This term arises from the expansion of (x + 8), representing the larger integer. The negative sign in -8x would suggest that we are dealing with integers that have a sum, rather than a difference, of 8, which contradicts the problem statement. Additionally, the positive +308 indicates that we haven't correctly accounted for the fact that the integers are negative and their product should result in a negative value when the equation is set to zero. Option C, x² + 8x + 308 = 0, shares a similar issue with the positive +308. This positive constant term implies that the product of the integers is being added to the equation, rather than subtracted, which is necessary to set the equation to zero and solve for x. In Option D, x² - 8x - 308 = 0, the correct negative constant term is present, but the -8x term is still problematic. As with Option B, this term suggests that the relationship between the integers is a subtraction of 8, rather than the addition implied by the problem statement. Therefore, each of these options contains critical errors in either the sign of the 8x term or the constant term, making them unsuitable for representing the given mathematical puzzle. This analysis emphasizes the importance of carefully translating word problems into algebraic equations, paying close attention to the signs and coefficients to ensure that the equation accurately reflects the conditions of the problem.

The Verdict: Option A is the Champion!

So, there you have it! Option A, $x^2 + 8x - 308 = 0$ , is the equation that correctly represents the relationship between our two negative integers. We cracked the code by translating the problem into algebraic language, expanding the equation, and comparing it to our options. Plus, we played detective and figured out why the other options just didn't fit the bill.

Remember, math problems like these aren't just about finding the right answer; they're about understanding the why behind the solution. By breaking down the problem, mapping out a strategy, and analyzing each step, we not only found the correct equation but also deepened our understanding of mathematical concepts. Keep practicing, keep exploring, and you'll be a math whiz in no time!