Simplify 1/(a+2b) + 2a/(a^2-4b^2): A Step-by-Step Guide

by Henrik Larsen 56 views

Hey guys! Let's dive into simplifying a pretty cool algebraic expression today. We're going to break down the expression 1/(a + 2b) + 2a/(a^2 - 4b^2) step by step so you can totally nail it. We'll cover everything from identifying key components to using algebraic identities like a pro. By the end of this, you'll not only know how to simplify this particular expression but also have some awesome tools for tackling similar problems. So, let's get started and make math a little less mysterious!

Understanding the Expression

Before we jump into the simplification process, let's make sure we really understand what we're looking at. Our main goal here is to combine two fractions: 1/(a + 2b) and 2a/(a^2 - 4b^2). Now, when you see fractions hanging out together like this, especially with different denominators (that's the bottom part of the fraction), our first instinct should be to find a common denominator. Why? Because we can't directly add or subtract fractions unless they have the same denominator. Think of it like trying to add apples and oranges – you need a common unit (like "fruit") to make sense of it. In this case, our common unit will be a common denominator.

The expression 1/(a + 2b) is a simple fraction where the denominator is a + 2b. Nothing too crazy here, right? But the second fraction, 2a/(a^2 - 4b^2), has a denominator that looks a bit more interesting: a^2 - 4b^2. This is where our algebra senses should start tingling! Does this denominator remind you of anything? It should! It's actually a difference of squares. Recognizing patterns like this is super important in algebra because it can help us simplify things a lot faster. The difference of squares pattern is a fundamental concept, and spotting it early can save you tons of time and effort. So, let's keep that in mind as we move forward.

Spotting the Difference of Squares

The difference of squares is a crucial algebraic identity that comes up all the time, so it's a fantastic one to have in your math toolkit. It states that x^2 - y^2 can be factored into (x + y)(x - y). This is like a magic trick that turns a subtraction of two squares into a product of two binomials. Now, let's see how this applies to our expression. In the denominator a^2 - 4b^2, we can recognize a^2 as the square of a and 4b^2 as the square of 2b. So, we can rewrite a^2 - 4b^2 as a^2 - (2b)^2. See how it fits the x^2 - y^2 pattern perfectly? This means we can factor it using the difference of squares identity.

Applying the identity, we get a^2 - (2b)^2 = (a + 2b)(a - 2b). This is a huge step forward! Why? Because now we can see that one of the factors, (a + 2b), is the same as the denominator of our first fraction. This is exactly what we need to find a common denominator. By factoring a^2 - 4b^2, we've made the connection between the two denominators clear and set ourselves up for simplification success. Remember, spotting these patterns is key, so keep an eye out for them in your algebraic adventures!

Finding a Common Denominator

Okay, now that we've successfully factored the denominator a^2 - 4b^2 into (a + 2b)(a - 2b), we're in a much better position to tackle the problem. Remember, our goal is to add the two fractions, 1/(a + 2b) and 2a/((a + 2b)(a - 2b)). To do this, we need a common denominator. Looking at the two denominators, (a + 2b) and (a + 2b)(a - 2b), it's pretty clear what our common denominator should be: (a + 2b)(a - 2b). This is because the second fraction already has this denominator, and the first fraction's denominator is a factor of it.

So, what do we do next? We need to make the first fraction have the common denominator. To do this, we multiply both the numerator (the top part) and the denominator (the bottom part) of the first fraction, 1/(a + 2b), by (a - 2b). Why do we multiply both the top and bottom by the same thing? Because it's like multiplying by 1 – it doesn't change the value of the fraction, only its appearance. This is a fundamental principle when working with fractions, and it's crucial for maintaining equality.

Multiplying to Get the Common Denominator

Let's walk through the multiplication step by step. We have the fraction 1/(a + 2b), and we want to multiply it by (a - 2b)/(a - 2b). This gives us:

(1 * (a - 2b)) / ((a + 2b) * (a - 2b)) = (a - 2b) / ((a + 2b)(a - 2b))

Now, look at that! The first fraction has the common denominator we were aiming for. The numerator is now (a - 2b), and the denominator is (a + 2b)(a - 2b), which is the same as a^2 - 4b^2. We haven't actually changed the value of the fraction; we've just rewritten it in a way that's more helpful for us. This step is essential because it allows us to combine the fractions in the next step. Once you get the hang of finding common denominators, you'll be able to simplify all sorts of complex expressions. So, let's move on and see how we can combine these fractions now that they're speaking the same denominator "language"!

Combining the Fractions

Alright, we've done the hard work of finding a common denominator. Now comes the fun part – combining the fractions! We have our two fractions:

(a - 2b) / ((a + 2b)(a - 2b)) and 2a / ((a + 2b)(a - 2b))

They both have the same denominator, which means we can go ahead and add their numerators. Remember, when you add fractions with a common denominator, you simply add the numerators and keep the denominator the same. It's like saying if you have 3 slices of a pizza and add 2 more slices, you now have 5 slices – the size of the slices (the denominator) stays the same.

So, let's add the numerators:

(a - 2b) + 2a

Adding the Numerators

Now, we need to simplify the numerator by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have a and 2a, which are like terms. We also have -2b, which doesn't have any like terms in the numerator. So, let's combine a and 2a:

a + 2a = 3a

Now, our simplified numerator looks like this:

3a - 2b

Great! We've simplified the numerator. Now, let's put it back over the common denominator to form our combined fraction:

(3a - 2b) / ((a + 2b)(a - 2b))

This is a significant step forward. We've successfully combined the two original fractions into a single fraction. But, we're not quite done yet! The next step is to see if we can simplify this fraction any further. Sometimes, we can cancel out common factors between the numerator and the denominator, which leads to an even simpler expression. So, let's investigate if there's anything we can cancel in our current fraction.

Simplifying the Result

We've arrived at the fraction (3a - 2b) / ((a + 2b)(a - 2b)). The big question now is: can we simplify this further? To do this, we need to look for common factors between the numerator, (3a - 2b), and the denominator, (a + 2b)(a - 2b). Remember, simplifying fractions often involves canceling out common factors, much like reducing a regular fraction like 4/6 to 2/3 by dividing both the numerator and denominator by 2.

Let's take a close look. The numerator is 3a - 2b. Can we factor this expression? Unfortunately, no. There's no common factor between 3a and 2b, and it doesn't fit any common factoring patterns like the difference of squares or perfect square trinomials. So, the numerator is as simple as it gets.

Now, let's examine the denominator, (a + 2b)(a - 2b). We already know that this is the factored form of a^2 - 4b^2. But does either of these factors, (a + 2b) or (a - 2b), match the numerator (3a - 2b)? Nope! There's no direct match.

Checking for Common Factors

Since we can't directly cancel any factors, we might be tempted to think we're done. However, it's always good to double-check. Sometimes, there might be a more subtle way to simplify. For instance, we could try expanding the denominator back to a^2 - 4b^2 and see if that gives us any new insights. But in this case, expanding the denominator doesn't reveal any further simplifications. The numerator (3a - 2b) simply doesn't share any factors with a^2 - 4b^2.

This is a crucial point: not every expression can be simplified further. Sometimes, the simplest form is exactly what you've got. In our case, the fraction (3a - 2b) / ((a + 2b)(a - 2b)) is indeed in its simplest form. We've done all we can to combine the fractions and reduce them, and there are no more simplifications to be made. So, we can confidently say that we've reached our final answer!

Final Simplified Expression

So, after all our hard work, we've successfully simplified the expression 1/(a + 2b) + 2a/(a^2 - 4b^2). We started by recognizing the difference of squares pattern, then found a common denominator, combined the fractions, and finally, checked for any further simplifications. And guess what? We nailed it!

The final simplified expression is:

(3a - 2b) / ((a + 2b)(a - 2b))

Or, if you prefer, you can write the denominator in its expanded form:

(3a - 2b) / (a^2 - 4b^2)

Both forms are perfectly valid and represent the same simplified expression. The key takeaway here is the process we followed. We didn't just blindly apply rules; we understood why each step was necessary. We recognized patterns, found common denominators, combined like terms, and checked for further simplifications. These are the skills that will help you conquer any algebraic expression that comes your way.

Key Takeaways

  • Recognize Patterns: Spotting the difference of squares (or other algebraic identities) is a game-changer.
  • Find Common Denominators: It's the golden rule for adding or subtracting fractions.
  • Combine Like Terms: Simplify numerators and denominators by grouping terms that are similar.
  • Check for Simplifications: Always look for common factors to cancel out.
  • Understand the Process: Don't just memorize steps; understand why you're doing them.

By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic challenges. So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!