Simplifying (2-a^2)/(a^2+a) + (3a+4)/(3a+3) A Step-by-Step Guide

Aug 7, 2025 by JurnalWarga.com 65 views

Simplifying and Discussing the Expression: (2-a^2)/(a^2+a) + (3a+4)/(3a+3)

Hey guys! Today, we're diving into a fun mathematical expression that involves fractions and variables. We're going to break down the expression (2-a^2)/(a2+a) + (3a+4)/(3a+3), simplify it, and discuss some important aspects related to it. Buckle up, it's going to be an interesting ride!

Understanding the Expression

Before we jump into simplification, let's take a good look at the expression we're dealing with:

(2-a^2)/(a2+a) + (3a+4)/(3a+3)

This expression consists of two fractions added together. Each fraction has a numerator and a denominator that involves the variable 'a'. Our goal is to simplify this expression into a more manageable form. The key to simplifying this expression lies in understanding how to work with fractions, factoring, and finding common denominators. So, let's start by taking a closer look at each part of the expression. First, let’s deeply understand each fraction separately. In the first fraction, (2-a^2)/(a2+a), we have a quadratic expression in the numerator and a quadratic expression in the denominator. Remember, guys, that quadratics often play a crucial role in simplifying expressions. We need to keep an eye out for ways to factorize them. In the second fraction, (3a+4)/(3a+3), both the numerator and the denominator are linear expressions. This means they involve 'a' to the power of 1. Sometimes, simplifying linear expressions involves finding common factors or just leaving them as is. When simplifying algebraic expressions, it's essential to identify the domain of the variable. The domain refers to the set of all possible values that the variable can take without making the expression undefined. In this case, we need to consider the denominators of the fractions. We know that a fraction is undefined when the denominator is equal to zero. Therefore, we need to find the values of 'a' that make the denominators zero and exclude them from the domain. Once we've simplified the expression, we can explore various aspects such as its behavior for different values of 'a', its graph, and any special properties it might have. This can give us a deeper understanding of the expression and its mathematical characteristics.

Step-by-Step Simplification

Now, let's roll up our sleeves and get our hands dirty with the simplification process! We'll break it down into manageable steps to make it easier to follow.

1. Factoring the Denominators

The first step in simplifying this expression is to factor the denominators of both fractions. Factoring helps us identify common factors and find a common denominator later on.

Denominator of the first fraction: a^2 + a. We can factor out an 'a' from both terms: a^2 + a = a(a + 1)
Denominator of the second fraction: 3a + 3. We can factor out a '3' from both terms: 3a + 3 = 3(a + 1)

So, our expression now looks like this:

(2 - a^2) / [a(a + 1)] + (3a + 4) / [3(a + 1)]

This is already looking better, right guys? Factoring the denominators helps us see the common factors more clearly.

2. Finding a Common Denominator

To add fractions, we need a common denominator. Looking at our factored denominators, we see that they have a common factor of (a + 1). The least common denominator (LCD) will be the product of all unique factors, each raised to the highest power it appears in any denominator. In this case, the LCD is 3a(a + 1).

Now, we need to rewrite each fraction with the LCD as its denominator.

First fraction: To get the LCD, we need to multiply the denominator a(a + 1) by 3. So, we also multiply the numerator by 3: [3(2 - a^2)] / [3a(a + 1)] = (6 - 3a^2) / [3a(a + 1)]
Second fraction: To get the LCD, we need to multiply the denominator 3(a + 1) by 'a'. So, we also multiply the numerator by 'a': [a(3a + 4)] / [3a(a + 1)] = (3a^2 + 4a) / [3a(a + 1)]

Now our expression looks like this:

(6 - 3a^2) / [3a(a + 1)] + (3a^2 + 4a) / [3a(a + 1)]

We're getting closer, guys! Now that we have a common denominator, we can add the fractions.

3. Adding the Fractions

Now that we have a common denominator, we can add the numerators:

[(6 - 3a^2) + (3a^2 + 4a)] / [3a(a + 1)]

Combine like terms in the numerator:

(6 - 3a^2 + 3a^2 + 4a) / [3a(a + 1)]

The -3a^2 and +3a^2 terms cancel each other out, leaving us with:

(6 + 4a) / [3a(a + 1)]

4. Further Simplification

We can simplify the numerator by factoring out a '2':

2(3 + 2a) / [3a(a + 1)]

Now, let's take a look at our simplified expression:

2(3 + 2a) / [3a(a + 1)]

Are we done? Well, we've simplified the expression as much as we can through factoring and combining terms. There are no more common factors to cancel out. So, this is our simplified form!

Discussion: Domain and Restrictions

Okay, so we've simplified the expression. Awesome! But we're not quite done yet. It's super important to discuss the domain and any restrictions on the variable 'a'. Remember, the domain is the set of all possible values that 'a' can take without making the expression undefined.

Identifying Restrictions

Fractions become undefined when the denominator is equal to zero. So, we need to find the values of 'a' that make our original denominators zero. Let's go back to the original expression:

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

We need to consider both denominators:

a^2 + a = 0 Factor out an 'a': a(a + 1) = 0 This gives us two solutions: a = 0 and a = -1
3a + 3 = 0 Factor out a '3': 3(a + 1) = 0 This gives us one solution: a = -1

So, we have two values that make the denominators zero: a = 0 and a = -1. These values are not allowed in the domain of our expression.

Stating the Domain

The domain of our expression is all real numbers except a = 0 and a = -1. We can write this in a few ways:

Set notation: {a | a ∈ ℝ, a ≠ 0, a ≠ -1}
Interval notation: (-∞, -1) ∪ (-1, 0) ∪ (0, ∞)

Understanding the domain is crucial because it tells us where our simplified expression is valid. If we try to plug in a = 0 or a = -1 into our simplified expression, we'll run into problems (like dividing by zero). Keep this in mind, guys!

Exploring the Simplified Expression

Now that we've simplified the expression and discussed its domain, let's take a moment to think about what our simplified form tells us:

2(3 + 2a) / [3a(a + 1)]

This expression is much easier to work with than our original one. We can use it to:

Evaluate the expression for different values of 'a' (as long as they're in the domain).
Analyze the behavior of the expression as 'a' gets very large or very small.
Graph the expression to visualize its properties.

For example, we could plug in a = 1 into our simplified expression:

2(3 + 2(1)) / [3(1)(1 + 1)] = 2(5) / [3(2)] = 10 / 6 = 5 / 3

So, when a = 1, the expression is equal to 5/3. Pretty cool, huh guys?

Conclusion

We've taken a deep dive into the expression (2-a^2)/(a2+a) + (3a+4)/(3a+3). We simplified it step-by-step, discussed the importance of the domain, and even explored some ways to use our simplified expression. Remember, simplifying algebraic expressions is a valuable skill in mathematics, and it often involves factoring, finding common denominators, and being mindful of restrictions. I hope you found this breakdown helpful and maybe even a little bit fun. Keep practicing, keep exploring, and most importantly, keep those math gears turning, guys! You've got this!