Factoring $x^2 - 2x - 80$ How-to Guide With Clear Steps

Factoring quadratic expressions is a fundamental skill in algebra, and it's super useful for solving equations, simplifying expressions, and understanding the behavior of functions. In this article, we'll dive deep into factoring the quadratic expression $x^2 - 2x - 80$. We'll walk through the process step-by-step, explore different techniques, and ultimately nail down the correct factorization. So, if you've ever felt a bit puzzled by factoring, don't worry – we've got you covered! Let's get started and make factoring feel like a breeze.

Understanding Quadratic Expressions

Before we jump into the nitty-gritty, let's get a handle on what quadratic expressions actually are. A quadratic expression is a polynomial that has a degree of two, meaning the highest power of the variable is 2. The general form of a quadratic expression is $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants (numbers), and 'x' is the variable. Think of it like this: you've got an $x^2$ term, an 'x' term, and a constant term hanging out together. Our expression, $x^2 - 2x - 80$, perfectly fits this mold, with $a = 1$, $b = -2$, and $c = -80$. Recognizing this form is the first step in mastering factoring. It's like knowing the shape of the puzzle piece before you try to fit it in! Now that we're all on the same page about what a quadratic expression looks like, we can move on to the exciting part: how to break it down.

When we talk about factoring a quadratic expression, what we're really trying to do is rewrite it as a product of two binomials. A binomial, in simple terms, is just an expression with two terms. So, we want to transform $x^2 - 2x - 80$ into something like $(x + p)(x + q)$, where 'p' and 'q' are numbers that we need to figure out. This process is like reverse-engineering the multiplication – we're going from the expanded form back to the factored form. Why do we do this? Well, factored form can make it much easier to solve equations and analyze the expression. It's like having a secret code that unlocks the expression's hidden properties. For example, if we set the factored expression equal to zero, we can quickly find the values of 'x' that make the expression zero, which are also known as the roots or zeros of the quadratic. Factoring is a powerful tool, and once you get the hang of it, you'll be amazed at how many problems it can help you solve. So, let's dive into the method for cracking this factoring puzzle!

In the context of factoring, the coefficients $a$, $b$, and $c$ play a crucial role. These constants are the keys to unlocking the factored form of the quadratic expression. For our example, $x^2 - 2x - 80$, we have $a = 1$, $b = -2$, and $c = -80$. The coefficient 'a' tells us about the leading term ($x^2$ in this case), 'b' is the coefficient of the linear term (the 'x' term), and 'c' is the constant term. These numbers are not just arbitrary; they hold valuable information about how the quadratic behaves and how it can be factored. The relationships between these coefficients are what guide our factoring process. For instance, the constant term 'c' gives us a clue about the possible constant terms in our binomial factors, while 'b' helps us refine our choices to ensure the middle term adds up correctly. So, keeping a close eye on these coefficients is essential for successful factoring. They are the ingredients in our algebraic recipe, and knowing how to use them will lead us to the perfect factorization!

The Factoring Method: Finding the Right Numbers

Okay, guys, let's get down to the nitty-gritty of actually factoring the expression $x^2 - 2x - 80$. The main idea behind factoring is to find two numbers that satisfy a specific set of conditions related to the coefficients of our quadratic expression. Remember those coefficients we talked about earlier? They're going to be our guide here. Specifically, we need to find two numbers, let's call them p and q, that do two things: First, p times q must equal the constant term, which is -80 in our case. This is the product condition. Second, p plus q must equal the coefficient of the 'x' term, which is -2. This is the sum condition. Finding these two magical numbers is like cracking a secret code, and once we have them, we're golden. So, how do we go about finding these numbers? Let's break it down.

Finding these magical numbers, p and q, is like a detective game. We need to look for clues and try different combinations until we hit the jackpot. A systematic way to do this is to start by listing the factor pairs of our constant term, -80. Factor pairs are simply pairs of numbers that multiply together to give us -80. Since we have a negative constant term, we know that one number in each pair must be positive, and the other must be negative. This narrows down our search a bit. Let's start listing those pairs: (1, -80), (-1, 80), (2, -40), (-2, 40), (4, -20), (-4, 20), (5, -16), (-5, 16), (8, -10), and (-8, 10). Whew, that's quite a few! Now, the next step is to check which of these pairs also adds up to our 'b' coefficient, which is -2. Remember, we need both the product and the sum conditions to be satisfied. So, we go through our list, adding the numbers in each pair. 1 + (-80) is -79, -1 + 80 is 79, and so on. As we go down the list, we're looking for that perfect pair that sums up to -2. Keep your eyes peeled, because the right pair is hiding in there! This methodical approach makes sure we don't miss any possibilities and helps us find the solution efficiently.

Now, let's take a closer look at our list and see which pair adds up to -2. We've got (1, -80), (-1, 80), (2, -40), (-2, 40), (4, -20), (-4, 20), (5, -16), (-5, 16), (8, -10), and (-8, 10). Adding the numbers in each pair, we quickly see that 8 + (-10) gives us -2. Bingo! We've found our magical numbers. So, p is 8, and q is -10 (or vice versa, it doesn't matter which way we assign them). These numbers are the key to unlocking the factored form of our quadratic expression. They're like the missing pieces of the puzzle, and now that we have them, we can put everything together. Remember, the hard work is often in finding the right numbers, but once you've got them, the rest is smooth sailing. So, now that we have our p and q, let's see how they fit into the factored form of the expression.

Constructing the Factored Form

Alright, we've nailed down the crucial numbers, 8 and -10, that satisfy our product and sum conditions. Now comes the fun part: putting them into the factored form! Remember, we're aiming to rewrite $x^2 - 2x - 80$ as a product of two binomials, like this: $(x + p)(x + q)$. We've found our p and q, so all we need to do is plug them in. Our numbers are 8 and -10, so we substitute them into the binomials. This gives us $(x + 8)(x - 10)$. Ta-da! We've constructed the factored form of the quadratic expression. It's like building a house – we had the individual pieces (the numbers), and now we've put them together to create the final structure (the factored form). But, before we celebrate too much, it's always a good idea to double-check our work. We want to make sure that $(x + 8)(x - 10)$ is indeed equivalent to our original expression, $x^2 - 2x - 80$. So, let's take a moment to verify our factorization.

To verify our factorization, we simply need to expand the factored form we just found, $(x + 8)(x - 10)$, and see if it matches our original expression, $x^2 - 2x - 80$. Expanding binomials involves multiplying each term in the first binomial by each term in the second binomial. This is often referred to as the FOIL method (First, Outer, Inner, Last), but you can think of it as just systematic distribution. Let's go through the steps: First, we multiply the 'x' in the first binomial by both terms in the second binomial: $x * x = x^2$ and $x * -10 = -10x$. Next, we multiply the '8' in the first binomial by both terms in the second binomial: $8 * x = 8x$ and $8 * -10 = -80$. Now, we have $x^2 - 10x + 8x - 80$. The last step is to combine like terms. We have two 'x' terms, -10x and 8x, which combine to give us -2x. So, our expanded expression is $x^2 - 2x - 80$. Guess what? It's exactly the same as our original expression! This confirms that our factorization is correct. We've successfully transformed the quadratic expression into its factored form, and we've proven that the two forms are equivalent. This verification step is like putting a seal of approval on our work, ensuring that we've got the right answer.

The Answer and Why It's Correct

After our journey through the factoring process, we've arrived at the factored form of the quadratic expression $x^2 - 2x - 80$. Our final answer is $(x + 8)(x - 10)$. This corresponds to option (d) in the original problem. But why is this the correct answer? We've already touched on this, but let's solidify our understanding. The key lies in the two numbers we found, 8 and -10. These numbers are special because they satisfy two crucial conditions: their product is equal to the constant term (-80), and their sum is equal to the coefficient of the 'x' term (-2). These conditions are the foundation of the factoring method we used. When we plug these numbers into the binomial form $(x + p)(x + q)$, we create an expression that, when expanded, will perfectly recreate our original quadratic expression. This is the magic of factoring – breaking down a complex expression into its simpler building blocks.

Our verification step further reinforces why $(x + 8)(x - 10)$ is the right answer. When we expanded this factored form, we meticulously multiplied each term and combined like terms, and we arrived back at our starting point: $x^2 - 2x - 80$. This expansion process is like reversing the factoring – we're taking the factored form and transforming it back into the original form. The fact that we ended up with the exact same expression proves that our factorization is accurate. It's like having a reversible recipe – we can go from the ingredients (the binomials) to the final dish (the quadratic expression), and we can also go from the final dish back to the ingredients. This two-way relationship is the hallmark of a correct factorization. So, we can confidently say that $(x + 8)(x - 10)$ is the factored form of $x^2 - 2x - 80$ because it satisfies the necessary conditions and can be verified through expansion. We've cracked the code!

Common Mistakes to Avoid

Factoring can be a bit tricky, and it's easy to stumble if you're not careful. Let's chat about some common mistakes that students often make so you can steer clear of them. One frequent error is getting the signs wrong. Remember, we need to find two numbers that multiply to give us the constant term and add up to give us the coefficient of the 'x' term. If you mix up the signs, you might end up with numbers that multiply correctly but don't add up correctly, or vice versa. For example, in our problem, we needed numbers that multiply to -80 and add to -2. If you mistakenly used -8 and 10, they would multiply to -80, but they add up to 2, not -2. So, always double-check those signs! Another common mistake is not considering all the factor pairs. It's tempting to stop searching once you find a pair that multiplies to the right number, but you need to make sure that pair also adds up to the correct value. We saw how listing out all the factor pairs systematically helped us find the right combination. So, take your time and be thorough in your search.

Another pitfall to watch out for is forgetting to double-check your work. We talked about how expanding the factored form is a great way to verify your answer, but many students skip this step. It's like baking a cake and not tasting it – you won't know if it's good until you try it! Expanding the factored form is your