Simplifying Rational Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of rational expressions. We'll be tackling operations and, more specifically, how to simplify them. Let's take a look at the following problem: $\frac{x+3}{2 x^2+x-6}-\frac{6 x-2}{3 x^2+5 x-2}$. Don't worry; it might look a bit intimidating at first glance, but breaking it down step-by-step will make it totally manageable. Think of it like a puzzle; we'll fit the pieces together, and you'll get a sense of accomplishment when we're done! This guide will walk you through the process, making sure you understand each step. Get ready to flex those math muscles!

Step 1: Factor Those Quadratics!

Alright, guys, the first step is always the most crucial: factoring those quadratic expressions. Remember, factoring is like finding the building blocks of our expressions. It helps us simplify things later on. So, we have two quadratics in our denominators: $2x^2 + x - 6$ and $3x^2 + 5x - 2$. Let's break them down individually. This will help us understand the structure and find the components to cancel out later on. We'll need to factor each quadratic expression in the denominator. Remember, the goal here is to rewrite the quadratics into products of simpler expressions. This will help us simplify later on.

For $2x^2 + x - 6$, we're looking for two binomials that, when multiplied, give us this expression. Through trial and error, or using methods like the 'ac method' (where you multiply the leading coefficient and the constant term, find factors that add up to the middle term's coefficient, and then rewrite and factor by grouping), we find that: $2x^2 + x - 6 = (2x - 3)(x + 2)$. If you're not feeling confident with factoring quadratics, take some time to review the techniques, as it is a foundational skill. Knowing how to factor is extremely important and can prevent you from making many mistakes. Keep practicing.

Next up, let's factor $3x^2 + 5x - 2$. Again, we're on the hunt for two binomials. After some factoring magic, we get: $3x^2 + 5x - 2 = (3x - 1)(x + 2)$. See, not too bad, right? Factoring these expressions opens the door to simplification. Now, it's time to rewrite our original expression, substituting the factored forms into our fractions. Understanding how to factor quadratics is a critical skill that will make your journey through algebra much smoother. Make sure you're comfortable with this step, as it's essential for simplifying our rational expressions. This is the key to simplifying and tackling more complex problems. Keep in mind that being able to factor is really the cornerstone of simplifying these expressions. With each quadratic expression successfully factored, we're one step closer to the final solution.

Step 2: Rewrite the Expression with Factored Denominators

Okay, we have done the first step, and we have now factored our quadratic equations. Now that we have factored the denominators, let's rewrite the original expression using the factored forms. This will make the next steps much easier to visualize and manage. The expression now looks like this: $\frac{x+3}{(2x-3)(x+2)} - \frac{6x-2}{(3x-1)(x+2)}$. Great! See how much cleaner it looks already? Rewriting the expression with the factored denominators is a visual win. This makes it easier to see common factors that might cancel out. The new representation gives us a clear view of the individual components of each fraction and sets us up for the next steps. So, let's rewrite our initial expression. Now that our denominators are factored, we have a better view of the components of our problem. This will help us see what common factors we have. Remember, our goal is to simplify the rational expressions, and rewriting with factored expressions is a step in the right direction. We're essentially swapping out the original quadratic denominators with their equivalent factored forms. Factoring transforms the complex problem into manageable parts. This makes the next steps much easier. It makes it easier to find common factors for us to use. This move helps us see what might cancel out and what might be left over. Now that we have done this, we can move on.

Step 3: Find the Least Common Denominator (LCD)

Alright, now it's time to find the Least Common Denominator (LCD). Think of the LCD as the smallest expression that both of our denominators can divide into evenly. To find the LCD, we look at all the unique factors in our denominators and take the highest power of each. Let's identify the factors in our expressions: $(2x - 3)$, $(x + 2)$, and $(3x - 1)$. Our LCD, therefore, is $(2x - 3)(x + 2)(3x - 1)$. This is the smallest expression that both denominators can divide into without leaving any remainders. To find the LCD, we need to look at the denominators of both fractions. Then, identify the factors of each denominator. This gives us the full scope of the denominator's components. The LCD is super important because it enables us to combine our fractions into a single term. So, we have to be very careful to get it right. This step is critical because it prepares us to combine the two fractions into a single, more manageable expression. The LCD ensures that when we combine our fractions, we are working with a common base. Remember, the LCD is like the 'universal language' that allows us to add or subtract fractions. It's the key to creating a combined fraction that will bring us closer to our solution. So, by identifying the LCD, we're setting the stage for the final calculation. Don't forget the LCD, as it's crucial for combining our fractions in the next steps! Now, we're ready to move on and combine the fractions.

Step 4: Rewrite Each Fraction with the LCD

Okay, guys, now that we have the LCD, we need to rewrite each fraction so that it has the LCD as its denominator. This might sound like a lot, but it's just like making sure all the fractions have the same 'language' before we add or subtract. Remember that the LCD is $(2x - 3)(x + 2)(3x - 1)$. Let's start with the first fraction, $\frac{x+3}{(2x-3)(x+2)}$. We have to multiply both the numerator and denominator by $(3x - 1)$. This gives us $\frac{(x+3)(3x-1)}{(2x-3)(x+2)(3x-1)}$.

Next, let's look at the second fraction, $\frac{6x-2}{(3x-1)(x+2)}$. We multiply both the numerator and denominator by $(2x - 3)$. This gives us $\frac{(6x-2)(2x-3)}{(3x-1)(x+2)(2x-3)}$.

So now, our expression looks like this: $\frac{(x+3)(3x-1)}{(2x-3)(x+2)(3x-1)} - \frac{(6x-2)(2x-3)}{(3x-1)(x+2)(2x-3)}$.

See how both fractions now share the same denominator? By rewriting each fraction with the LCD, we are ensuring that we can combine them correctly. Always remember to multiply both the numerator and denominator by the same expression, as this ensures that you're not changing the value of the fraction. It's like multiplying by 1, which doesn't alter the equation but allows you to adjust the appearance. It's like giving each fraction a 'makeover' so that they all look the same underneath. The rewriting step might seem tedious, but it is a necessary step to prepare our fractions for the next phase. Remember that the goal here is to combine the fractions and simplify, so this step will pay off soon.

Step 5: Combine the Fractions

Alright, we've got everything set up perfectly. Now it's time to combine the fractions. Because both fractions now share the same denominator, $(2x - 3)(x + 2)(3x - 1)$, we can combine the numerators. Our expression now looks like this: $\frac{(x+3)(3x-1)-(6x-2)(2x-3)}{(2x-3)(x+2)(3x-1)}$. See how we just put the numerators together over the common denominator? This is what it's all about! We've now transformed our original problem into a single fraction. Combining the fractions is all about consolidating everything under one common denominator, which we've worked hard to establish. Now that the fractions are together, we can simplify the numerator. We're one step closer to completing this problem. This step gets us closer to our final answer! So, let's continue.

Step 6: Simplify the Numerator

Now for the fun part: simplifying the numerator. This involves expanding the products in the numerator, combining like terms, and generally cleaning things up. We have $(x+3)(3x-1)-(6x-2)(2x-3)$ in our numerator. Let's expand those products: $(x+3)(3x-1) = 3x^2 + 8x - 3$, and $(6x-2)(2x-3) = 12x^2 - 22x + 6$. Now, subtract the second expanded expression from the first: $(3x^2 + 8x - 3) - (12x^2 - 22x + 6)$. This simplifies to $-9x^2 + 30x - 9$. So, our expression now looks like this: $\frac{-9x^2 + 30x - 9}{(2x-3)(x+2)(3x-1)}$.

At this point, we want to simplify the numerator. Make sure you distribute the negative sign correctly when subtracting. Simplifying the numerator is all about organizing the terms and combining like terms. Expanding those products and simplifying is what gives us our new numerator. Remember, precision is key here. Make sure you don't lose any terms, and take your time. This step requires careful attention to detail because a small mistake in the expansion or subtraction can throw off the entire solution. The goal of this step is to make sure our numerator is as simple as possible. We're getting closer to the end. Keep going, you've got this.

Step 7: Factor the Numerator (Again!) and Look for Simplification

Guess what, guys? We're going to factor again! This time, we'll factor the numerator, which is $-9x^2 + 30x - 9$. You can start by factoring out a common factor of $-3$: $-3(3x^2 - 10x + 3)$. Now, let's factor the quadratic expression inside the parentheses. We're looking for two numbers that multiply to 9 and add up to -10. Those numbers are -1 and -9, so the expression factors to $-3(3x - 1)(x - 3)$. So our expression is now: $\frac{-3(3x-1)(x-3)}{(2x-3)(x+2)(3x-1)}$. Now is the time to look for potential simplifications. We're trying to see if we can cancel anything out. The common factor of $(3x-1)$ appears in both the numerator and the denominator. We can cancel these out. This leaves us with: $\frac{-3(x-3)}{(2x-3)(x+2)}$.

Factoring the numerator again and then looking for simplifications is a key step to simplify this expression. By factoring, you might find common factors between the numerator and the denominator, which helps you simplify it. Factoring the numerator again is a key move because it allows us to identify any common factors that we can cancel out with the denominator. Now we are ready for the last step. And we are almost at the finish line. Keep going.

Step 8: Final Answer

After all that hard work, we've simplified the expression. Our final answer is $\frac{-3(x-3)}{(2x-3)(x+2)}$. Congrats, you made it! Always remember to factor the numerator and denominator completely, and then look for common factors that can be canceled out. This is how we reduce rational expressions to their simplest form. Keep practicing and you'll become a master of these expressions. Remember, there are always many ways to approach a math problem. The more problems you work, the more comfortable you will become. So, keep at it. You got this. And now, you've successfully simplified the rational expression!