Solving Equations: Finding A+b+c In A Rational Expression

Hey math enthusiasts! Let's dive into a cool algebra problem today. We're going to solve an equation involving rational expressions and figure out the value of a + b + c. Sounds fun, right? Don't worry, it's easier than it looks! We'll break it down step by step, so you can totally nail it. So, let's get started!

Understanding the Problem: The Core Equation

Alright guys, the main dish of our problem is this equation: $\frac{3x + 1}{x - 1} - \frac{4 - x}{x + 2} = \frac{ax^2 + bx + c}{(x - 1)(x + 2)}$

Our mission, should we choose to accept it (and we do!), is to find the values of a, b, and c, and then add them all up. This might seem like a lot, but trust me, with a little algebra magic, we'll crack this code. The key is to simplify the left side of the equation until it looks like the right side. That will allow us to see the relationship between the coefficients of the polynomial in the numerator. Basically, we need to make the left side look like a single fraction with the same denominator as the right side. Once we've done that, the numerators will have to be equal. That is when we can easily find the value of a, b, and c. It's like a puzzle, and we're the puzzle masters!

To make life easier, we need to deal with the fractions on the left side. Notice that the right side has a common denominator of (x - 1)(x + 2). So, that’s our game plan: we'll rewrite the left side with the same common denominator. This involves a little bit of algebraic manipulation, but don't worry, it’s not rocket science. It's about finding equivalent fractions. Basically, we will adjust the fractions to get a common denominator, then we can combine them. Remember, equivalent fractions are just fractions that represent the same value, but they have different numerators and denominators. Understanding this concept is really important if you want to be able to answer questions like these in the future!

Let’s begin by addressing the first fraction on the left. To get the common denominator of (x - 1)(x + 2), we'll multiply the first fraction's numerator and denominator by (x + 2). This is the same as multiplying by 1 (since anything divided by itself equals 1), so we're not changing the value of the fraction, just its appearance. Next, for the second fraction, we'll multiply the numerator and denominator by (x - 1). See? We're just making sure both fractions have the same denominator, which is crucial for combining them. Keep in mind that we're basically doing this to set up the subtraction. Now, let’s get into the details of the steps involved in solving this problem.

Step-by-Step Solution: Unveiling the Values

Okay, guys, let’s get our hands dirty and solve this equation step-by-step. First, we need to rewrite each fraction on the left side so they have the common denominator of (x - 1)(x + 2). This might seem tricky at first, but it's really just a matter of multiplying the numerator and denominator of each fraction by the appropriate factor. It’s like a little algebra dance. We'll start with the first fraction, (3x + 1) / (x - 1). We'll multiply both the numerator and denominator by (x + 2). This gives us (3x + 1)(x + 2) / ((x - 1)(x + 2)). Now for the second fraction, (4 - x) / (x + 2), we will multiply the numerator and denominator by (x - 1). This will result in (4 - x)(x - 1) / ((x - 1)(x + 2)). Now the two fractions have the same denominator, as planned. After this, we can subtract the second fraction from the first, which is the whole point of this operation.

After we've found the equivalent fractions with the common denominator, we can now subtract them. We combine the numerators over the common denominator. Remember to be super careful with the signs! This is where mistakes often happen. We want to take (3x + 1)(x + 2) and subtract (4 - x)(x - 1). Make sure to distribute that negative sign correctly across all the terms in the second fraction's numerator. This becomes extremely important, as the negative sign can often be overlooked, and it can really mess up your result! It's like a secret agent, always ready to change the equation. We’ll carefully expand and simplify the numerator. So, we'll end up with something like (3x^2 + 7x + 2) - (4x - 4 - x^2 + x) . Then, combining like terms, we can simplify this expression.

Let’s now focus on simplifying the numerators. We need to expand the products in the numerator. This requires a little bit of basic algebra, and we can do it! For the first product, (3x + 1)(x + 2), we use the distributive property (or the FOIL method, if you’re into that). This means multiplying each term in the first set of parentheses by each term in the second set. So, we get 3x*x + 3x*2 + 1*x + 1*2, which simplifies to 3x^2 + 6x + x + 2. Then combine the like terms, so the expanded form is 3x^2 + 7x + 2. Doing the same for the second product, (4 - x)(x - 1), we get 4*x + 4*(-1) - x*x -x*(-1), which simplifies to 4x - 4 - x^2 + x. Combining the like terms, this simplifies to -x^2 + 5x - 4. After having done that, you are almost there!

Finally, we'll combine the terms, remember to subtract the entire second numerator. Thus the equation becomes, (3x^2 + 7x + 2) - (-x^2 + 5x - 4) / ((x - 1)(x + 2)) . Now simplify. Pay super close attention to the signs here, since subtracting a negative is the same as adding. (3x^2 + 7x + 2) + (x^2 - 5x + 4). Combining like terms, we get 4x^2 + 2x + 6. So the left side of the equation simplifies to (4x^2 + 2x + 6) / ((x - 1)(x + 2)). At this point, the left side of the equation should look like the right side, so we can determine the values of a, b, and c! Comparing the numerators of both sides gives us ax^2 + bx + c = 4x^2 + 2x + 6. From this, we can easily see that a = 4, b = 2, and c = 6. Now that you know the values of a, b, and c, calculating the value of a+b+c is just a piece of cake.

Calculating a + b + c: The Grand Finale

Alright, folks, we're at the finish line! We have the values of a, b, and c, and now we need to calculate a + b + c. This is the easiest part. We know that a = 4, b = 2, and c = 6. So, let’s plug them in and add them up: 4 + 2 + 6 = 12. That's it! We have our answer. We went from a complex-looking equation to a simple solution. We successfully found the values and used them to solve this problem! This is the power of persistence and basic algebra. So, the correct answer is C. 12. Great job, guys!

Checking Your Work: Double-Checking the Solution

It's always a good idea to double-check your work, and the easiest way to do this is to substitute your values of a, b, and c back into the original equation and verify the identity. This will ensure that all the computations are correct, and will help you develop confidence in your math skills. In our case, the right side of the equation becomes (4x^2 + 2x + 6) / ((x - 1)(x + 2)). Now compare it with our simplified version of the left side. If you did everything correctly, you'll see that both sides are identical, confirming that our answer is correct. This is called a sanity check, and it's something that will become useful as you advance further in math. Remember, even the best mathematicians make mistakes, so always double-check your work to catch those little errors. This is how you develop the confidence to take on any mathematical problem!

Conclusion: Mastering Rational Expressions

And that, my friends, is how you solve this type of equation. We started with a complicated rational expression and, with a few clever algebraic moves, found the values of a, b, and c, and then found their sum. We broke it down into small, manageable steps. Remember the importance of finding a common denominator, expanding and simplifying expressions, and paying close attention to signs. These are your secret weapons in the world of algebra. By practicing these techniques, you'll become a pro at solving rational equations. Keep practicing, keep learning, and keep having fun with math! You got this! Also, don't forget to subscribe for more math problems and solutions! Thanks for joining me today. See you in the next one!

Key Takeaways:

• Find Common Denominators: This is the first and most important step in adding or subtracting rational expressions.
• Simplify Numerators: Expand and combine like terms carefully.
• Pay Attention to Signs: A common source of errors is incorrect sign handling.
• Double-Check Your Work: Always verify your solution by substituting the values back into the original equation.
• Practice Makes Perfect: The more you practice, the easier it will become!