Simplify (2x²-10x-28)/(6x) * 6/(x-7): Step-by-Step Guide

Hey guys! Ever stared at a mathematical expression that looks like it belongs in a monster movie? Don't worry, we've all been there. Today, we're going to tackle one of those beasts and break it down into its simplest form. Get ready to roll up your sleeves and dive into the world of algebra! We'll be working with the expression (2x²-10x-28)/(6x) * 6/(x-7). It might look intimidating, but trust me, it's like a puzzle waiting to be solved. So, let's grab our mathematical magnifying glasses and get started!

The Challenge: A Complex Expression

So, the expression we're going to simplify is this: $\frac{2 x^2-10 x-28}{6 x} \times \frac{6}{x-7}$. It looks like a jumble of numbers, variables, and fractions, right? But don't let it scare you! We're going to take it step by step and turn it into something much simpler. Our main goal here is to simplify this expression as much as possible. This means we want to reduce it to its most basic form, where there are no more common factors to cancel out. To do this, we'll use our knowledge of factoring, fractions, and algebraic manipulation. Think of it like decluttering your room – we're getting rid of all the unnecessary stuff to reveal the beautiful simplicity underneath. The expression (2x²-10x-28)/(6x) * 6/(x-7) might seem daunting at first glance, but with the right tools and techniques, it can be tamed. This expression involves a combination of quadratic expressions, fractions, and multiplication. Our task is to simplify this complex expression by factoring, canceling common terms, and reducing it to its most basic form. This involves several key algebraic concepts, including factoring quadratic expressions, multiplying fractions, and simplifying rational expressions. By mastering these concepts, we can confidently tackle similar problems in the future and gain a deeper understanding of algebraic manipulation. Remember, simplification isn't just about getting the right answer; it's about understanding the underlying structure and relationships within the expression. It's like peeling back the layers of an onion to reveal the core – the simplest, most fundamental form. So, let's dive in and start peeling!

Step-by-Step Simplification

Factoring the Quadratic Expression

First things first, let's focus on that quadratic expression in the numerator: 2x² - 10x - 28. Remember factoring from your algebra days? We need to find two numbers that multiply to give us the product of the leading coefficient (2) and the constant term (-28), which is -56, and add up to the middle coefficient (-10). Think of it like a detective game, where we're searching for the perfect pair of numbers to unlock the mystery of the quadratic. After a bit of number sleuthing, we find that -14 and 4 fit the bill perfectly. These two numbers, -14 and 4, are the key to unlocking the factors of our quadratic expression. They allow us to rewrite the middle term (-10x) as a combination of -14x and +4x, which will then enable us to factor by grouping. Factoring by grouping is a powerful technique that allows us to break down a complex expression into simpler, more manageable parts. It's like taking apart a machine to see how each piece fits together. We can rewrite the quadratic expression as 2x² - 14x + 4x - 28. Now, we can group the first two terms and the last two terms: (2x² - 14x) + (4x - 28). From the first group, we can factor out a 2x, and from the second group, we can factor out a 4. This gives us 2x(x - 7) + 4(x - 7). Notice that we now have a common factor of (x - 7) in both terms! This is a crucial step, as it allows us to further simplify the expression. By factoring out the (x - 7), we are essentially collapsing the two terms into a single, more compact form. We can now factor out the (x - 7), which gives us (x - 7)(2x + 4). Voila! We've successfully factored the quadratic expression. This factored form, (x - 7)(2x + 4), is equivalent to the original quadratic expression, but it's much more useful for simplification. It's like transforming a tangled mess of yarn into a neatly wound ball – much easier to work with!

Rewriting and Simplifying the Expression

Now that we've factored the quadratic, let's rewrite our original expression: $\frac{(x-7)(2x+4)}{6x} \times \frac{6}{x-7}$. See how things are starting to look clearer? It's like the fog is lifting, and we can finally see the path ahead. We've replaced the quadratic expression with its factored form, which makes it easier to spot common factors that we can cancel out. This is a crucial step in simplifying rational expressions, as it allows us to eliminate terms that appear in both the numerator and the denominator. It's like finding matching socks in your laundry pile – you can pair them up and remove them from the pile, making the remaining socks easier to sort. The next step is to identify common factors in the numerator and the denominator. We have a (x - 7) in both the numerator and the denominator, and we also have a 6 in both the numerator and the denominator. These common factors are like the repeating patterns in a kaleidoscope – they can be simplified and removed to reveal a simpler, more elegant design. Canceling these common factors is like dividing both the top and bottom of a fraction by the same number – it doesn't change the value of the fraction, but it makes it simpler to work with. So, let's cancel out the (x - 7) terms and the 6 terms. This leaves us with $\frac{2x+4}{x}$. We're getting closer to our simplified expression! This simplified expression, $\frac{2x+4}{x}$, is much cleaner and easier to understand than the original expression. It's like removing the unnecessary words from a sentence to make it clearer and more concise. However, we're not quite done yet! We can still simplify this expression further by factoring out a common factor from the numerator.

Final Simplification

Notice that in the numerator, 2x + 4, both terms have a common factor of 2. Let's factor that out! Factoring out the 2 is like finding a common ingredient in two different recipes – you can use it to simplify both recipes and make them more efficient. Factoring out the 2 gives us 2(x + 2). Now our expression looks like this: $\frac{2(x+2)}{x}$. This expression is equivalent to the previous one, but it's in a more factored form, which makes it easier to see if we can simplify further. It's like rearranging the furniture in your room to create more space and make it more functional. Now, let's put it all together. Our simplified expression is $\frac{2(x+2)}{x}$, which can also be written as $\frac{2x+4}{x}$. This is the simplest form of our original expression! We've successfully navigated the maze of algebraic manipulation and arrived at our destination. We've transformed a complex expression into a simple, elegant form that is much easier to understand and work with. This journey through simplification has not only given us the answer, but also a deeper appreciation for the power and beauty of algebra. We've seen how factoring, canceling common terms, and simplifying rational expressions can transform seemingly complex expressions into something much more manageable. This is a skill that will serve us well in future mathematical adventures. So, congratulations on reaching the final step! You've successfully simplified a challenging expression and expanded your algebraic toolkit.

The Answer and Why

The simplest form of the expression $\frac{2 x^2-10 x-28}{6 x} \times \frac{6}{x-7}$ is A. $\frac{2 x+4}{x}$. We arrived at this answer by factoring the quadratic expression, canceling common factors, and simplifying. Remember, practice makes perfect! The more you work with these types of problems, the more confident you'll become in your algebra skills. Keep exploring, keep learning, and keep simplifying! This journey of mathematical exploration is full of exciting discoveries and challenges. By embracing these challenges and working through them step by step, we not only arrive at the right answers but also develop a deeper understanding of the underlying concepts. So, don't be afraid to tackle complex problems – each one is an opportunity to learn and grow. And remember, the beauty of mathematics lies not just in the answers, but in the process of finding them.

Choosing the Right Path: Why Other Options Don't Fit

Let's quickly look at why the other options are incorrect. This is a great way to reinforce our understanding and make sure we're on the right track. It's like double-checking your route on a map to make sure you haven't taken a wrong turn. By analyzing why the other options are incorrect, we solidify our understanding of the correct method and avoid making similar mistakes in the future. It's like learning from your past experiences to improve your future performance. Option B, 6, is incorrect because we didn't account for the variables in the expression. We can't just cancel everything out and end up with a constant. Option C, $\frac{12 x+24}{6 x}$, is a simplified form, but not the simplest form. We can still factor out a common factor of 6 from the numerator and denominator. Option D, $\frac{2 x^2-10 x-168}{6 x^2-42 x}$, is what you might get if you multiplied the numerators and denominators without factoring first. This makes the expression more complex, not simpler. Understanding why these options are incorrect helps us to appreciate the importance of following the correct steps in simplification. It's like understanding the consequences of making a wrong decision – it motivates us to make the right choices in the future. So, by analyzing the incorrect options, we strengthen our understanding of the correct approach and build our confidence in our problem-solving skills.

Level Up Your Math Skills

Simplifying expressions like this is a fundamental skill in algebra. It's like learning the alphabet in order to read and write – it's a building block for more advanced concepts. Mastering simplification techniques will open doors to a whole new world of mathematical possibilities. The more comfortable you are with simplifying expressions, the easier it will be to tackle more complex problems in algebra, calculus, and beyond. It's like learning to ride a bike – once you've mastered the basics, you can explore new trails and go on exciting adventures. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. Remember, math is not just about numbers and equations; it's about logical thinking, problem-solving, and the beauty of abstract concepts. It's a language that allows us to describe and understand the world around us. So, embrace the challenge, enjoy the journey, and keep leveling up your math skills!