Solving Fractions A Step-by-Step Guide To 5/(5x+2) - 2/(9x^2-4)

Hey guys! Today, we're diving deep into the world of fractions, specifically tackling the problem: ${\frac{5}{5x+2} - \frac{2}{9x^2-4}}$. This type of problem often appears in algebra and calculus, so mastering it is super important for your math journey. We'll break it down step-by-step, making sure you understand not just the how but also the why behind each step. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into solving, let's understand what we're dealing with. We have two fractions: ${\frac{5}{5x+2}}$ and ${\frac{2}{9x^2-4}}$. Our goal is to subtract the second fraction from the first. To do this, we need a common denominator. Think of it like trying to add or subtract slices of different-sized pizzas – you need to cut them into the same size slices first!

Why is a common denominator so crucial? Well, when fractions have the same denominator, it means we're dealing with the same "unit" or "size of the pieces." This allows us to directly compare and combine the numerators (the top parts of the fractions). Without a common denominator, we're essentially trying to add apples and oranges – they're just not compatible.

In this particular problem, the denominators are ${5x+2}$ and ${9x^2-4}$. Notice anything special about the second denominator? It's a difference of squares! Recognizing patterns like this is a key skill in algebra, and it will help us simplify the problem significantly. We'll factor this expression in the next step, which will bring us closer to finding that common denominator we need.

So, to recap, we're subtracting fractions, we need a common denominator, and we've spotted a difference of squares pattern. Feeling good? Let's move on!

Factoring the Denominators

The first step in finding a common denominator is to factor each denominator completely. This helps us identify the common and unique factors we'll need. Our denominators are ${5x+2}$ and ${9x^2-4}$.

The first denominator, ${5x+2}$, is a linear expression and cannot be factored further. It's in its simplest form. So, we can just leave it as is.

The second denominator, ${9x^2-4}$, is where things get interesting. As we noted earlier, this is a difference of squares. Remember the difference of squares pattern? It's ${a^2 - b^2 = (a + b)(a - b)}$. This is a super useful pattern to memorize, guys, because it pops up all the time in algebra.

In our case, we can see that ${9x^2}$ is ${(3x)^2}$ and ${4}$ is ${2^2}$. So, we can apply the difference of squares pattern:

${9x^2 - 4 = (3x + 2)(3x - 2)}$

Now we have factored both denominators:

• ${5x + 2}$ remains as ${5x + 2}$
• ${9x^2 - 4}$ factors into ${(3x + 2)(3x - 2)}$

Factoring is like breaking down a number into its prime factors. It helps us see the building blocks of the expression. In this case, it reveals the factors that will help us find our common denominator. Next up, we'll identify the least common denominator (LCD).

Identifying the Least Common Denominator (LCD)

Okay, we've factored the denominators, which is a huge step! Now, let's find the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. Think of it as the smallest "common ground" for our fractions.

We have the factored denominators:

• ${5x + 2}$
• ${(3x + 2)(3x - 2)}$

To find the LCD, we take each unique factor and its highest power that appears in either denominator. This ensures that the LCD is divisible by both original denominators. It's like making sure we have enough ingredients to make both recipes.

Let's break it down:

1. The factor ${(5x + 2)}$ appears in the first denominator. So, we include it in the LCD.
2. The factors ${(3x + 2)}$ and ${(3x - 2)}$ appear in the second denominator. So, we include both of them in the LCD.

Therefore, the LCD is:

${(5x + 2)(3x + 2)(3x - 2)}$

Notice that we didn't just multiply the original denominators together. That would have given us a common denominator, but not necessarily the least common denominator. Using the LCD keeps our expressions simpler and easier to work with. It's like using the most efficient route to get to your destination.

Now that we have the LCD, we can rewrite each fraction with this new denominator. This is the key step that allows us to actually subtract the fractions. Let's see how it's done!

Rewriting Fractions with the LCD

Alright, we've found the LCD: ${(5x + 2)(3x + 2)(3x - 2)}$. Now, we need to rewrite each fraction with this denominator. This involves multiplying both the numerator and the denominator of each fraction by the factors that are missing from its original denominator. It's like scaling up the fractions to fit the same "size slice."

Let's start with the first fraction: ${\frac{5}{5x+2}}$

Its denominator is ${5x + 2}$. To get the LCD, we need to multiply it by ${(3x + 2)(3x - 2)}$. Remember, we have to multiply both the numerator and the denominator by the same expression to keep the fraction equivalent. It's like multiplying by 1 – it changes the form but not the value.

So, we have:

${\frac{5}{5x+2} \cdot \frac{(3x+2)(3x-2)}{(3x+2)(3x-2)} = \frac{5(3x+2)(3x-2)}{(5x+2)(3x+2)(3x-2)}}$

Now, let's move on to the second fraction: ${\frac{2}{9x^2-4}}$. We already factored the denominator as ${(3x + 2)(3x - 2)}$.

To get the LCD, we need to multiply this denominator by ${(5x + 2)}$. Again, we multiply both the numerator and the denominator:

${\frac{2}{(3x+2)(3x-2)} \cdot \frac{5x+2}{5x+2} = \frac{2(5x+2)}{(5x+2)(3x+2)(3x-2)}}$

Now, both fractions have the same denominator – the LCD! This means we're finally ready to subtract them. We're in the home stretch, guys!

To recap, we've rewritten the fractions so they have a common denominator. This involved multiplying the numerator and denominator of each fraction by the missing factors. Next, we'll combine the numerators and simplify the expression.

Combining and Simplifying

Great job, guys! We've successfully rewritten both fractions with the LCD: ${(5x + 2)(3x + 2)(3x - 2)}$. Now comes the fun part – combining the numerators and simplifying the expression. This is where we put everything together and get our final answer.

We have:

${\frac{5(3x+2)(3x-2)}{(5x+2)(3x+2)(3x-2)} - \frac{2(5x+2)}{(5x+2)(3x+2)(3x-2)}}$

Since the denominators are the same, we can combine the numerators:

${\frac{5(3x+2)(3x-2) - 2(5x+2)}{(5x+2)(3x+2)(3x-2)}}$

Now, let's simplify the numerator. First, we'll expand the terms. Remember, ${(3x + 2)(3x - 2)}$ is a difference of squares, so it expands to ${9x^2 - 4}$:

${\frac{5(9x^2-4) - 2(5x+2)}{(5x+2)(3x+2)(3x-2)}}$

Next, we distribute the constants:

${\frac{45x^2 - 20 - 10x - 4}{(5x+2)(3x+2)(3x-2)}}$

Now, combine like terms in the numerator:

${\frac{45x^2 - 10x - 24}{(5x+2)(3x+2)(3x-2)}}$

Now, let's see if we can factor the numerator. This can sometimes lead to further simplification by canceling common factors with the denominator. The numerator is a quadratic expression, so we can try to factor it. After attempting to factor the numerator (45x^2 - 10x - 24), it does not factor easily, and for the purpose of this solution, we'll assume it doesn't have common factors with the denominator.

So, our simplified expression is:

${\frac{45x^2 - 10x - 24}{(5x+2)(3x+2)(3x-2)}}$

We can also leave the denominator in its factored form or expand it, depending on the context or instructions of the problem. However, leaving it factored often makes it easier to spot potential cancellations or further simplifications in future steps.

And there you have it! We've successfully subtracted the fractions and simplified the result. You guys rock!

Final Thoughts and Takeaways

Wow, guys, we did it! We tackled a complex fraction subtraction problem and came out victorious. Let's recap the key steps we took:

1. Understood the problem: We identified the need for a common denominator to subtract fractions.
2. Factored the denominators: We recognized and applied the difference of squares pattern.
3. Identified the LCD: We found the smallest expression divisible by both denominators.
4. Rewrote fractions with the LCD: We multiplied numerators and denominators to get equivalent fractions.
5. Combined and simplified: We subtracted the numerators and simplified the expression.

The biggest takeaway here is that breaking down complex problems into smaller, manageable steps makes them much easier to solve. Each step, like factoring or finding the LCD, has its own logic and purpose. When you understand these individual steps, you can tackle even the most daunting problems with confidence.

Remember these key concepts:

• Common Denominators: Essential for adding or subtracting fractions.
• Factoring: Unlocking hidden structures and simplifying expressions.
• LCD: The most efficient common denominator.
• Simplifying: Presenting your answer in its cleanest form.

This type of problem is fundamental in algebra and calculus, so mastering it will definitely pay off in your future math studies. Keep practicing, and don't be afraid to ask questions. You've got this!

I hope this comprehensive guide has helped you understand how to solve ${\frac{5}{5x+2} - \frac{2}{9x^2-4}}$. Keep up the great work, and I'll see you in the next math adventure!