Simplifying Rational Expressions Finding Equivalent Form Of X/(x^2-1) + 4/(x^2-6x-7)
Hey guys! Today, we are diving into the fascinating world of rational expressions. Our mission is to simplify a complex expression and find its equivalent form. Specifically, we're tackling the sum of two fractions: x/(x^2-1) and 4/(x^2-6x-7). This is a common type of problem you might encounter in algebra, and mastering it is crucial for more advanced math topics. So, let's break it down step by step, making sure everyone understands the process. We'll cover everything from factoring to finding common denominators and simplifying the final result. By the end of this guide, you'll be able to confidently handle similar problems. Remember, the key to success in math is practice, so don't hesitate to work through additional examples on your own. Let's get started and make math a little less daunting and a lot more fun!
Breaking Down the Problem: Factoring the Denominators
So, when we are faced with adding fractions, especially algebraic ones, the first thing we need to do, guys, is to factor the denominators. Why? Because finding the least common denominator (LCD) becomes a whole lot easier when we know the factors. Think of it like organizing your closet – it's much simpler to find what you need when everything is neatly arranged. Let's take a closer look at our denominators:
- x^2 - 1: This looks familiar, right? It's the difference of squares! Remember the formula: a^2 - b^2 = (a + b)(a - b). Applying this, we can factor x^2 - 1 into (x + 1)(x - 1). See? Much simpler already.
- x^2 - 6x - 7: Okay, this one is a quadratic expression. We need to find two numbers that multiply to -7 and add up to -6. After a bit of thought, we'll find that -7 and +1 fit the bill perfectly. So, we can factor x^2 - 6x - 7 into (x - 7)(x + 1). Nice!
Now that we've factored both denominators, our original expression looks a bit different and a lot more manageable:
x / [(x + 1)(x - 1)] + 4 / [(x - 7)(x + 1)]
Factoring the denominators is like laying the foundation for a building. It's the essential first step that makes the rest of the process smooth and straightforward. Now that we've got this down, we can move on to the next crucial step: finding the least common denominator. Keep up the great work, and remember, each step we take gets us closer to the solution!
Finding the Least Common Denominator (LCD)
Alright, guys, now that we've successfully factored the denominators, the next big step is to find the least common denominator (LCD). Think of the LCD as the common ground where our fractions can finally meet and combine. It's the smallest expression that each denominator can divide into evenly. So, how do we find this magical LCD? It’s simpler than you might think!
Look at the factored denominators we found earlier:
- (x + 1)(x - 1)
- (x - 7)(x + 1)
The LCD needs to include all the unique factors present in either denominator, and if a factor appears more than once in any single denominator, we take the highest power of that factor. In our case, we have the factors (x + 1), (x - 1), and (x - 7). Notice that (x + 1) appears in both denominators, but only to the first power, so we just include it once in our LCD.
Therefore, the LCD is the product of all these unique factors:
LCD = (x + 1)(x - 1)(x - 7)
See? It’s like building a bridge that connects all the denominators. Now that we have our LCD, we can rewrite each fraction with this common denominator. This is a crucial step because it allows us to add the fractions together. Remember, you can only add fractions if they have the same denominator – it's like trying to add apples and oranges directly; you need a common unit (like “fruits”) to make it work. So, let's move on to the next step and see how we rewrite our fractions with the LCD. You're doing great – keep going!
Rewriting Fractions with the LCD
Okay, everyone, we've got our LCD, which is (x + 1)(x - 1)(x - 7). Now comes the part where we make sure both fractions are speaking the same language – that is, having the same denominator. This involves rewriting each fraction using the LCD we just found. It might sound a bit tricky, but I promise, it's totally manageable if we take it step by step.
Let's start with the first fraction:
x / [(x + 1)(x - 1)]
We need to figure out what we're missing in the denominator to make it the LCD. Looking at our LCD, we see that we're missing the factor (x - 7). So, we need to multiply both the numerator and the denominator of this fraction by (x - 7). Remember, it's crucial to multiply both the top and bottom to keep the fraction equivalent to its original form. It’s like multiplying by 1 – you change the appearance, but not the value.
So, we get:
[x * (x - 7)] / [(x + 1)(x - 1)(x - 7)]
Which simplifies to:
[x(x - 7)] / [(x + 1)(x - 1)(x - 7)]
Now, let's do the same for the second fraction:
4 / [(x - 7)(x + 1)]
Here, we're missing the factor (x - 1) in the denominator. So, we multiply both the numerator and the denominator by (x - 1):
[4 * (x - 1)] / [(x - 7)(x + 1)(x - 1)]
Which simplifies to:
[4(x - 1)] / [(x + 1)(x - 1)(x - 7)]
Now, check it out! Both fractions have the same denominator – our LCD! We've successfully rewritten our fractions, and we're all set to combine them. Give yourself a pat on the back; this is a significant step. Next up, we'll add those fractions together and see what we get. Keep your momentum going; you're doing fantastic!
Combining the Fractions
Alright, guys, this is where the magic happens! We've done the groundwork – factoring, finding the LCD, and rewriting the fractions. Now, we get to the main event: combining the fractions. Since both fractions now have the same denominator, adding them is as straightforward as adding regular fractions.
We have:
[x(x - 7)] / [(x + 1)(x - 1)(x - 7)] + [4(x - 1)] / [(x + 1)(x - 1)(x - 7)]
When fractions have the same denominator, you simply add the numerators and keep the denominator the same. It’s like adding slices of the same size pizza – you just count the slices!
So, we add the numerators:
x(x - 7) + 4(x - 1)
And keep the common denominator:
(x + 1)(x - 1)(x - 7)
This gives us a combined fraction:
[x(x - 7) + 4(x - 1)] / [(x + 1)(x - 1)(x - 7)]
We're not quite done yet, though. The next step is to simplify the numerator by expanding and combining like terms. This is like tidying up after the party – we want to make sure everything is in its place and as neat as possible. So, let's roll up our sleeves and simplify that numerator. You're doing great – just a bit more to go!
Simplifying the Numerator
Okay, everyone, we've got our combined fraction:
[x(x - 7) + 4(x - 1)] / [(x + 1)(x - 1)(x - 7)]
Now, it's time to simplify the numerator. This means we need to expand the terms and then combine any like terms. Think of it as decluttering – we want to make the expression as clean and simple as possible.
Let's start by expanding the terms in the numerator:
- x(x - 7) expands to x^2 - 7x
- 4(x - 1) expands to 4x - 4
So, our numerator now looks like this:
x^2 - 7x + 4x - 4
Next, we combine the like terms. In this case, we have -7x and +4x, which combine to -3x. So, the simplified numerator is:
x^2 - 3x - 4
Now, our fraction looks like this:
(x^2 - 3x - 4) / [(x + 1)(x - 1)(x - 7)]
But wait, we're not quite done yet! We can actually factor the numerator further. This is like adding the final touch to a masterpiece – we want to make sure it's perfect. Let's see if we can factor that quadratic expression. You're doing awesome – keep pushing through to the finish line!
Factoring the Simplified Numerator
Alright, team, we've simplified the numerator to x^2 - 3x - 4, and now it's time to see if we can factor it further. Factoring, as we've seen, is a powerful tool for simplifying expressions. It's like finding the hidden code within the numbers.
We need to find two numbers that multiply to -4 and add up to -3. After a little bit of thinking, we can see that -4 and +1 fit the bill perfectly. So, we can factor x^2 - 3x - 4 into:
(x - 4)(x + 1)
Now, our fraction looks like this:
[(x - 4)(x + 1)] / [(x + 1)(x - 1)(x - 7)]
Do you see what's coming next? We've got a common factor in both the numerator and the denominator! This is our chance to simplify even further. It's like finding a shortcut that makes the journey even easier. So, let's go ahead and cancel out that common factor. You're doing such a fantastic job – we're almost at the finish line!
Canceling Common Factors and Final Answer
Okay, everyone, the moment we've been working towards is here! We've got our fraction:
[(x - 4)(x + 1)] / [(x + 1)(x - 1)(x - 7)]
And we've spotted a common factor: (x + 1) appears in both the numerator and the denominator. This means we can cancel it out! Canceling common factors is like cutting away the unnecessary bits to reveal the true essence of the expression.
When we cancel out (x + 1), we're left with:
(x - 4) / [(x - 1)(x - 7)]
And there you have it! This is the simplest form of our original expression. We've successfully added the two fractions and simplified the result. It was quite a journey, but we made it together. You've shown amazing persistence and skill throughout this process.
So, the final answer is:
(x - 4) / [(x - 1)(x - 7)]
Give yourselves a huge round of applause! You've conquered a challenging problem in rational expressions. Remember, the key to mastering math is practice and patience. Keep working hard, and you'll be able to tackle any math problem that comes your way. Great job, everyone!