Multiplying Fractions: Solve (2a-7)/a * (3a^2)/(2a^2-11a+14)

by Andrew McMorgan 61 views

Hey guys! Today, we're diving into a super cool problem involving the multiplication of algebraic fractions. This might sound intimidating, but trust me, it's totally manageable once you break it down. We're going to tackle the question: What is the simplified result of multiplying the fractions (2a-7)/a and (3a2)/(2a2-11a+14)? So, grab your pencils, and let's get started!

Understanding the Basics of Multiplying Fractions

Before we jump into the algebraic stuff, let's quickly refresh the basics of multiplying fractions. Remember, when you multiply fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, if you have 1/2 multiplied by 2/3, you get (1 * 2) / (2 * 3), which simplifies to 2/6, and further simplifies to 1/3. Easy peasy, right? Now, let's see how this applies to our algebraic fractions.

The beauty of algebra is that it allows us to work with variables and expressions just like we work with regular numbers. When we're dealing with algebraic fractions, we're essentially doing the same thing as with numerical fractions – multiplying numerators and denominators. However, we also need to keep an eye out for opportunities to simplify, especially by factoring and canceling common factors. This is where things get a little more interesting, and where we can really show off our math skills.

In the context of our problem, understanding the basic principle of multiplying numerators and denominators is the first step. We have two fractions, each with its own numerator and denominator, and our goal is to combine them into a single, simplified fraction. To do this, we'll multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. But before we rush into multiplying everything out, it's often a good idea to see if we can simplify anything first. This can save us a lot of work in the long run and make the problem much easier to handle. Think of it as looking for shortcuts in a maze – the fewer twists and turns, the better!

Step-by-Step Solution

1. Write down the problem:

The first step is always to clearly write down what we're trying to solve. This helps us keep everything organized and reduces the chance of making a silly mistake. Our problem is:

(2a - 7) / a * (3a^2) / (2a^2 - 11a + 14)

2. Factor, if possible:

Factoring is a super important technique in algebra, and it's going to be our best friend here. We need to look at each part of the fractions and see if we can break them down into simpler pieces. The numerator (2a - 7) looks pretty simple already, so we'll leave that for now. The denominator 'a' is also as simple as it gets. But what about the other fraction? The numerator (3a^2) is straightforward, but the denominator (2a^2 - 11a + 14) looks like it might be factorable.

Let's focus on factoring the quadratic expression 2a^2 - 11a + 14. This is where we need to think about two numbers that multiply to give us the product of the leading coefficient (2) and the constant term (14), which is 28, and add up to the middle coefficient (-11). After a bit of thought, we can see that -4 and -7 fit the bill perfectly. So, we can rewrite the quadratic as:

2a^2 - 4a - 7a + 14

Now, we'll use factoring by grouping. We group the first two terms and the last two terms:

(2a^2 - 4a) + (-7a + 14)

Next, we factor out the greatest common factor from each group:

2a(a - 2) - 7(a - 2)

Notice that we now have a common factor of (a - 2) in both terms. We can factor this out:

(2a - 7)(a - 2)

Awesome! We've successfully factored the quadratic expression. Now our problem looks like this:

(2a - 7) / a * (3a^2) / [(2a - 7)(a - 2)]

3. Multiply the fractions:

Now that we've factored everything we can, it's time to multiply the fractions. Remember, we multiply the numerators together and the denominators together:

[(2a - 7) * (3a^2)] / [a * (2a - 7)(a - 2)]

This gives us:

(3a^2 * (2a - 7)) / (a * (2a - 7) * (a - 2))

4. Simplify by canceling common factors:

This is where the magic happens! We can cancel out any factors that appear in both the numerator and the denominator. Notice that we have (2a - 7) in both the numerator and the denominator, so we can cancel those out. Also, we have 'a' in the denominator and 'a^2' in the numerator. We can cancel out one 'a' from each, leaving us with just 'a' in the numerator:

(3a * (2a - 7)) / (a * (2a - 7) * (a - 2))

After canceling the (2a - 7) terms, we have:

(3a^2) / (a * (a - 2))

And after canceling one 'a' from the numerator and denominator, we get:

3a / (a - 2)

5. Final Answer:

So, the simplified result of multiplying the fractions is:

3a / (a - 2)

Common Mistakes to Avoid

When you're working on problems like this, it's super easy to make little mistakes that can throw off your whole answer. But don't worry, we're gonna go over some of the most common pitfalls so you can steer clear of them!

One biggie is forgetting to factor completely. It's like trying to build a house without all the right pieces – you might get somewhere, but it won't be as solid as it could be. Always double-check to see if you can factor any further, especially with quadratic expressions. Trust me, taking that extra minute to factor can save you a lot of headaches later on.

Another sneaky mistake is only factoring part of an expression. Imagine you're untangling a string of lights, but you only untangle half of it – you're still left with a mess! Make sure you factor the entire expression, not just a piece of it. This means looking at all the terms and seeing if there are any common factors that can be pulled out.

And hey, we've all been there – rushing through a problem and making a simple arithmetic error. It's like accidentally adding 2 and 2 and getting 5. It happens! That's why it's always a good idea to take your time and double-check your work, especially when you're dealing with negative signs or tricky numbers. A little extra care can go a long way in making sure you get the right answer.

Practice Problems

Okay, guys, now it's your turn to shine! To really nail this concept, practice is key. Here are a few problems similar to the one we just solved. Try working through them on your own, and don't peek at the answers until you've given it your best shot!

  1. Simplify: (x + 2) / (x^2 - 9) * (x + 3) / (x^2 + 4x + 4)
  2. What is the product of (4b - 8) / (b^2) and (3b^3) / (2b - 4)?
  3. Multiply and simplify: (y^2 - 4) / (y^2 + 5y + 6) * (y + 3) / (y - 2)

Remember, the goal here isn't just to get the right answer, but to understand the process. So, take your time, show your work, and think about each step as you go. If you get stuck, that's totally okay! Go back and review the example we worked through together, or ask a friend or teacher for help. The more you practice, the more confident you'll become in your algebra skills.

Conclusion

And there you have it! We've successfully multiplied and simplified algebraic fractions. Remember, the key is to factor, multiply, and then simplify by canceling common factors. Keep practicing, and you'll become a pro at this in no time. Keep rocking those math skills!