Factoring 18w² + 21w - 9 Completely A Step-by-Step Guide

Hey everyone! Today, we're diving into a common algebra problem: factoring quadratic expressions. Specifically, we're going to completely factor the expression 18w² + 21w - 9. Factoring might seem daunting at first, but trust me, with a systematic approach, it becomes quite manageable. Let’s break it down step-by-step, so even if you're just starting with algebra, you'll be able to follow along. We'll cover everything from identifying the greatest common factor to using different factoring techniques. This is a crucial skill in algebra, as it's used in solving equations, simplifying expressions, and much more. So, buckle up and let's get started!

Understanding the Basics of Factoring

Before we jump into the specifics of our expression, let's quickly recap what factoring is all about. At its core, factoring is the reverse process of expanding. When we expand, we multiply terms together; when we factor, we break an expression down into its multiplicative components. For instance, if we expand 3(x + 2), we get 3x + 6. Factoring, in this case, would mean starting with 3x + 6 and breaking it back down into 3(x + 2). In the realm of quadratic expressions like ours (18w² + 21w - 9), factoring helps us rewrite the expression in a simpler, often more useful form. This can be particularly handy when we're trying to solve equations or simplify complex algebraic problems. Think of it as taking a puzzle apart to see how it fits together, then putting it back together in a new, often more organized way. Factoring is also fundamental for understanding the roots (or solutions) of a quadratic equation, as it helps us identify the values of 'w' that make the expression equal to zero. So, mastering factoring is not just about manipulating expressions; it's about gaining deeper insights into the behavior of equations and functions. We’ll be using this concept throughout our solution today, ensuring that every step makes sense and helps you build a solid foundation in algebra.

Step 1: Finding the Greatest Common Factor (GCF)

Whenever you're faced with a factoring problem, the first thing you should always check for is the greatest common factor (GCF). This is the largest number or expression that divides evenly into all the terms in your expression. In our case, we have 18w² + 21w - 9. Looking at the coefficients (18, 21, and -9), we can see that they are all divisible by 3. So, 3 is a common factor. Now, let's examine the variable terms. We have w² and w, but the constant term -9 doesn't have any 'w's. This means that 'w' itself is not a common factor across all terms. Therefore, the GCF for our expression is simply 3. Factoring out the GCF is crucial because it simplifies the expression, making it easier to factor further. It’s like tidying up your workspace before tackling a big project – it just makes everything more manageable. By dividing each term by 3, we reduce the complexity of the numbers we're dealing with, which in turn reduces the chances of making mistakes in the subsequent steps. This initial step is a fundamental principle in factoring, and it's one you should always apply before attempting more advanced techniques. It's all about making the problem as simple as possible before diving into the more intricate details.

Step 2: Factoring Out the GCF

Now that we've identified the GCF as 3, let's factor it out from the expression 18w² + 21w - 9. To do this, we divide each term in the expression by 3 and then write the result in parentheses, with the GCF (which is 3) outside the parentheses. So, when we divide 18w² by 3, we get 6w². When we divide 21w by 3, we get 7w. And when we divide -9 by 3, we get -3. Putting it all together, we have 3(6w² + 7w - 3). Notice how factoring out the GCF has made the numbers smaller and the expression inside the parentheses simpler. This is a key benefit of factoring out the GCF – it sets us up for easier factoring in the next steps. Think of it like this: if you're trying to solve a complex puzzle, it helps to first group the pieces that obviously belong together. Factoring out the GCF is like that first step of grouping, making the rest of the puzzle-solving process smoother. At this point, we've successfully simplified the original expression, and we're now ready to tackle the quadratic expression inside the parentheses. The next step involves factoring this quadratic, which will lead us to the completely factored form of our original expression. Remember, each step we take is a building block in the overall process, and understanding each step thoroughly is crucial for mastering factoring.

Step 3: Factoring the Quadratic Expression

Alright, we've got our expression down to 3(6w² + 7w - 3). Now, the challenge is to factor the quadratic expression inside the parentheses: 6w² + 7w - 3. There are several methods we can use, but one of the most common is the ac method. This method involves a bit of detective work with numbers, but it's quite effective once you get the hang of it. First, we identify 'a', 'b', and 'c' in our quadratic expression, which is in the standard form of aw² + bw + c. In our case, a = 6, b = 7, and c = -3. The 'ac method' gets its name because the first step is to multiply 'a' and 'c', so we calculate 6 * -3, which gives us -18. Next, we need to find two numbers that multiply to -18 and add up to 'b', which is 7. This is where the number detective work comes in! We need to think of pairs of factors of -18. After a little thought, we can see that the numbers 9 and -2 fit the bill, because 9 * -2 = -18 and 9 + (-2) = 7. These numbers are key to breaking down the middle term (7w) into two separate terms, which will allow us to factor by grouping. Factoring quadratics can feel like a puzzle, but the 'ac method' provides a structured way to find the right pieces. It's all about breaking down the problem into smaller, more manageable steps, and that's exactly what we're doing here. Once we've found these crucial numbers, we're ready to rewrite the quadratic expression and move on to the next phase of factoring.

Step 4: Rewriting the Middle Term and Factoring by Grouping

Now that we've found the magic numbers (9 and -2) from the 'ac method', we can rewrite our quadratic expression. Remember, we're working with 6w² + 7w - 3, and we're going to split the middle term (7w) using our numbers. So, we rewrite 7w as 9w - 2w. This gives us a new expression: 6w² + 9w - 2w - 3. Notice that we haven't changed the value of the expression; we've just rewritten it in a way that allows us to use factoring by grouping. Factoring by grouping involves pairing the terms and factoring out the GCF from each pair. It's a neat technique that turns a four-term expression into a product of two binomials. First, let's group the first two terms and the last two terms: (6w² + 9w) + (-2w - 3). From the first group (6w² + 9w), we can factor out 3w, which leaves us with 3w(2w + 3). From the second group (-2w - 3), we can factor out -1 (be mindful of the negative sign!), which gives us -1(2w + 3). Now, our expression looks like this: 3w(2w + 3) - 1(2w + 3). Do you see the common factor? Both terms have (2w + 3) in them! This is the key to the final step of factoring by grouping. By rewriting the middle term and applying this grouping technique, we're essentially reorganizing the expression to reveal its factored form. It’s like rearranging a deck of cards to see the patterns and combinations more clearly. This method is a powerful tool in your factoring arsenal, and it's especially useful for quadratics that aren't easily factored by simple trial and error.

Step 5: Final Factoring Step

We've made it to the final step of factoring the quadratic expression! We're at the point where we have 3w(2w + 3) - 1(2w + 3). As we noticed earlier, both terms have a common factor of (2w + 3). So, we can factor out (2w + 3) from the entire expression. When we factor out (2w + 3), we're left with 3w from the first term and -1 from the second term. This gives us (2w + 3)(3w - 1). Remember, we had the GCF of 3 that we factored out at the very beginning. We need to bring that back into the picture to get the completely factored expression. So, we multiply the entire expression by 3, which gives us 3(2w + 3)(3w - 1). And there you have it! We've successfully factored the quadratic expression. This final step is like putting the last piece of the puzzle in place. You've done the hard work of breaking down the expression and rearranging it, and now you can see the complete picture: the fully factored form. Factoring out the common binomial factor is a crucial skill in algebra, and it's one that you'll use time and time again. By mastering this technique, you'll be able to simplify complex expressions, solve equations, and tackle a wide range of algebraic problems with confidence. Now, let’s take a step back and appreciate what we've accomplished – we've taken a quadratic expression and broken it down into its simplest factors!

Conclusion: The Completely Factored Expression

So, guys, after all our hard work, we've successfully factored the expression 18w² + 21w - 9 completely! We started by identifying and factoring out the greatest common factor (GCF), which was 3. This gave us 3(6w² + 7w - 3). Then, we tackled the quadratic expression inside the parentheses using the 'ac method' and factoring by grouping. We found the magic numbers, rewrote the middle term, grouped the terms, and factored out the common binomial. Finally, we arrived at the completely factored form: 3(2w + 3)(3w - 1). This result is the equivalent of our original expression, but it's broken down into its simplest multiplicative components. Factoring is a fundamental skill in algebra, and it's essential for solving equations, simplifying expressions, and understanding the behavior of functions. By mastering these techniques, you'll be well-equipped to tackle more advanced algebraic problems. Remember, the key to factoring is to take it step-by-step, always looking for the GCF first, and then applying appropriate factoring methods for the remaining expression. Whether it's the 'ac method', factoring by grouping, or other techniques, each method has its place in your toolkit. Keep practicing, and you'll become more confident and proficient in factoring. And who knows, you might even start to enjoy the challenge of breaking down complex expressions into their simpler forms!