How To Factor X^2 - 2x - 80 A Step-by-Step Guide

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Factoring quadratic expressions can seem daunting at first, but with a systematic approach, it becomes a manageable task. In this comprehensive guide, we'll break down the process of factoring the quadratic expression $x^2 - 2x - 80$, providing a clear, step-by-step explanation to help you understand the underlying concepts. By the end of this article, you'll not only be able to factor this specific expression but also gain the skills to tackle similar quadratic problems with confidence. Let's dive in and unlock the secrets of factoring!

Understanding Quadratic Expressions

Before we jump into the specifics of factoring $x^2 - 2x - 80$, let's first establish a solid understanding of what quadratic expressions are and why they're important in algebra. At its core, a quadratic expression is a polynomial expression of degree two. This means that the highest power of the variable (usually 'x') is two. The general form of a quadratic expression is $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Understanding quadratic expressions is crucial in various areas of mathematics, including solving equations, graphing parabolas, and even in real-world applications like physics and engineering. When we delve into factoring quadratics, we're essentially trying to reverse the process of multiplication. Think of it like this: if we multiply two binomials, such as $(x + m)$ and $(x + n)$, we get a quadratic expression. Factoring, then, is the process of finding those two binomials that multiply together to give us the original quadratic expression.

Why is this important? Well, factoring allows us to simplify complex expressions, solve quadratic equations, and gain insights into the behavior of quadratic functions. In essence, it's a fundamental skill that opens doors to more advanced mathematical concepts. For instance, when solving quadratic equations, factoring can help us find the roots or solutions – the values of 'x' that make the equation equal to zero. These roots have significant meaning in various contexts, such as determining the points where a parabola intersects the x-axis. Moreover, factoring is not just a theoretical exercise; it has practical applications in fields like physics, where quadratic equations are used to model projectile motion, and in engineering, where they are used in structural design. So, whether you're a student grappling with algebra or someone interested in the broader applications of mathematics, mastering the art of factoring quadratic expressions is a valuable investment. Let's move forward and see how we can apply these concepts to our specific problem.

Identifying the Components of $x^2 - 2x - 80$

Now, let's focus on the specific quadratic expression we're going to factor: $x^2 - 2x - 80$. The first step in factoring any quadratic expression is to identify its components. Remember the general form of a quadratic expression: $ax^2 + bx + c$. In our case, we need to determine the values of 'a', 'b', and 'c'. This is crucial because these coefficients will guide our factoring process. Looking at $x^2 - 2x - 80$, we can see that:

• The coefficient 'a' is the number in front of the $x^2$ term. Here, since there's no visible number, it's understood to be 1. So, $a = 1$.
• The coefficient 'b' is the number in front of the 'x' term. In this expression, it's -2. So, $b = -2$.
• The constant term 'c' is the number without any 'x' attached. In this case, it's -80. So, $c = -80$.

It's essential to pay close attention to the signs of 'b' and 'c', as they play a significant role in determining the factors. A negative 'b' indicates that the sum of the factors we're looking for will be negative, while a negative 'c' indicates that the factors will have opposite signs (one positive and one negative). In our expression, the negative 'b' and 'c' tell us that we need to find two numbers that have a difference of 2 (the absolute value of 'b') and multiply to -80.

Identifying these components correctly is like laying the foundation for a building. If you get this step right, the rest of the factoring process becomes much smoother. It's a simple yet vital step that shouldn't be overlooked. Many factoring errors stem from misidentifying 'a', 'b', or 'c'. So, take your time, double-check your work, and ensure you've accurately pinpointed these coefficients before moving on. Now that we know our 'a', 'b', and 'c', we're ready to move on to the next step: finding the factors that will help us break down the expression.

Finding the Factors of -80 That Sum to -2

Now comes the core of the factoring process: finding the right factors. This step involves a bit of detective work, where we need to identify two numbers that satisfy two crucial conditions: they must multiply to give us the value of 'c' (which is -80 in our case), and they must add up to the value of 'b' (which is -2). This is where your number sense and knowledge of multiplication tables come in handy.

Let's start by listing the factor pairs of 80 (ignoring the negative sign for now):

• 1 and 80
• 2 and 40
• 4 and 20
• 5 and 16
• 8 and 10

Remember, since 'c' is negative (-80), one of the factors must be positive, and the other must be negative. Also, since 'b' is negative (-2), the larger of the two factors should be negative. This narrows down our search considerably. Looking at the list, we need a pair of factors that have a difference of 2 (because they will have opposite signs, and their sum needs to be -2).

Can you spot the pair? It's 8 and 10! If we make 10 negative, we have -10 and 8. Let's check if these factors work:

• 8 * -10 = -80 (This matches our 'c' value)
• 8 + (-10) = -2 (This matches our 'b' value)

Bingo! We've found our factors. The numbers 8 and -10 satisfy both conditions, which means we're on the right track to factoring our quadratic expression. This step might feel like a bit of trial and error at first, but with practice, you'll become more adept at quickly identifying the correct factors. The key is to be systematic in your approach, listing out factor pairs and considering the signs carefully. Now that we've successfully identified our factors, we're ready to use them to rewrite our quadratic expression in a factored form. Let's see how this is done in the next step.

Rewriting the Expression Using the Factors

With the factors 8 and -10 in hand, we're now ready to rewrite our quadratic expression, $x^2 - 2x - 80$. This step is crucial because it bridges the gap between identifying the factors and expressing the quadratic in its factored form. The core idea here is to split the middle term (-2x) into two terms using the factors we found. This might seem a bit counterintuitive at first, but it's a clever technique that allows us to group terms and ultimately factor the expression.

So, instead of writing -2x, we'll write 8x - 10x. Notice that 8x - 10x is indeed equal to -2x, so we're not changing the value of the expression, just its appearance. Our expression now looks like this:

$x^2 + 8x - 10x - 80$

We've essentially taken the original three-term expression and turned it into a four-term expression. This might seem like we've made things more complicated, but trust me, this is a necessary step towards factoring. Now, the beauty of this rewriting becomes clear: we can group the terms in pairs and factor out the greatest common factor (GCF) from each pair.

This technique is called factoring by grouping, and it's a powerful tool in our factoring arsenal. It allows us to break down a complex expression into smaller, more manageable parts. By rewriting the middle term using our factors, we've set the stage for this grouping process. In the next step, we'll delve into the actual factoring by grouping and see how it leads us to the final factored form of our quadratic expression. So, stick with me, and let's unlock the next piece of the puzzle!

Factoring by Grouping

Now that we've rewritten our expression as $x^2 + 8x - 10x - 80$, it's time to put the factoring by grouping technique into action. This method relies on strategically pairing terms and extracting their greatest common factors (GCFs). It's like a mathematical assembly line, where each step brings us closer to the final factored form. Let's break it down:

First, we group the first two terms and the last two terms together:

$(x^2 + 8x) + (-10x - 80)$

Notice that we've kept the signs consistent within the parentheses. This is crucial for the next step. Now, we'll factor out the GCF from each group.

For the first group, $(x^2 + 8x)$, the GCF is 'x'. Factoring out 'x', we get:

x(x + 8)

For the second group, $(-10x - 80)$, the GCF is -10. Factoring out -10, we get:

-10(x + 8)

Notice anything interesting? Both groups now have a common factor of $(x + 8)$. This is not a coincidence! If you've chosen the correct factors and performed the grouping correctly, you should always end up with a common binomial factor at this stage. Our expression now looks like this:

x(x + 8) - 10(x + 8)

Now, we can factor out the common binomial factor $(x + 8)$ from the entire expression. This is the final step in the factoring by grouping process. When we factor out $(x + 8)$, we're left with 'x' from the first term and -10 from the second term. So, we get:

(x + 8)(x - 10)

And there you have it! We've successfully factored our quadratic expression by grouping. This technique might seem a bit intricate at first, but with practice, it becomes a powerful tool in your factoring toolkit. The key is to rewrite the middle term using the correct factors, group the terms strategically, and factor out the GCFs. Now that we've arrived at the factored form, let's take a moment to verify our result and ensure we've got it right.

Verifying the Solution

We've arrived at our factored form: $(x + 8)(x - 10)$. But how can we be sure that this is the correct factorization of $x^2 - 2x - 80$? The best way to verify our solution is to multiply the factors back together and see if we get the original expression. This process is essentially the reverse of factoring, and it's a great way to check your work and build confidence in your factoring skills.

Let's multiply $(x + 8)$ and $(x - 10)$ using the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last):

• First: x * x = $x^2$
• Outer: x * -10 = -10x
• Inner: 8 * x = 8x
• Last: 8 * -10 = -80

Now, let's combine these terms:

$x^2$ - 10x + 8x - 80

Combining the like terms (-10x and 8x), we get:

$x^2$ - 2x - 80

And there it is! This is exactly the original expression we started with. This confirms that our factorization is correct. Verifying your solution is a crucial step in the factoring process. It's like proofreading a piece of writing – it helps you catch any errors and ensures that your final answer is accurate. By multiplying the factors back together, you're essentially creating a mathematical feedback loop that reinforces your understanding of factoring.

So, always take the time to verify your solutions, especially when you're first learning how to factor. It's a small investment of time that can save you from making mistakes and solidify your grasp of the concepts. Now that we've verified our solution, we can confidently say that we've successfully factored the quadratic expression $x^2 - 2x - 80$. But before we conclude, let's take a look at the multiple-choice options provided and select the correct answer.

Selecting the Correct Option

Now that we've successfully factored the quadratic expression $x^2 - 2x - 80$ and verified our solution, it's time to select the correct option from the given choices. This step is straightforward, as we've already done the heavy lifting of factoring. Our factored form is $(x + 8)(x - 10)$. Let's look at the options:

a) $(x - 8)(x + 10)$ b) $(x + 6)(x - 1)$ c) $(x + 3)(x + 6)$ d) $(x + 8)(x - 10)$

By comparing our factored form with the options, we can clearly see that option d) $(x + 8)(x - 10)$ matches our solution. Therefore, option d) is the correct answer. This final step highlights the importance of having a solid understanding of the factoring process. Without it, you might be tempted to guess or choose an incorrect option. But by following our step-by-step guide, you were able to confidently arrive at the correct answer.

Selecting the correct option is not just about getting the right answer; it's also about demonstrating your understanding of the underlying concepts. It shows that you've not only memorized a process but also grasped the logic behind it. So, congratulations! You've successfully factored the quadratic expression and chosen the correct option. But our journey doesn't end here. Let's take a moment to recap the key steps and reinforce our understanding of factoring quadratic expressions.

Conclusion: Mastering Quadratic Factoring

We've reached the end of our journey to factor the quadratic expression $x^2 - 2x - 80$, and what a journey it has been! We've covered a lot of ground, from understanding the basics of quadratic expressions to the intricacies of factoring by grouping. Let's take a moment to recap the key steps we've learned:

1. Understanding Quadratic Expressions: We started by defining what quadratic expressions are and why they're important in algebra.
2. Identifying the Components: We learned how to identify the coefficients 'a', 'b', and 'c' in the quadratic expression.
3. Finding the Factors: We tackled the core of factoring by finding two numbers that multiply to 'c' and add up to 'b'.
4. Rewriting the Expression: We split the middle term using the factors we found, setting the stage for factoring by grouping.
5. Factoring by Grouping: We grouped terms, factored out GCFs, and ultimately arrived at the factored form.
6. Verifying the Solution: We multiplied the factors back together to ensure we got the original expression.
7. Selecting the Correct Option: Finally, we confidently chose the correct answer from the given options.

By following these steps, you can approach any quadratic factoring problem with a clear and systematic strategy. Remember, factoring is not just about finding the right answer; it's about developing a deep understanding of algebraic principles. It's a skill that will serve you well in various areas of mathematics and beyond. So, don't be discouraged if you find it challenging at first. Practice makes perfect, and the more you factor, the more confident and proficient you'll become.

Keep exploring, keep learning, and keep factoring! The world of mathematics is full of fascinating challenges, and with the right tools and techniques, you can conquer them all. Now, go forth and factor with confidence!