Simplifying Algebraic Expressions: A Step-by-Step Guide

Let's dive into the world of algebraic expressions! The goal is to simplify and manipulate them, making them easier to understand and work with. In this case, our focus is on an expression involving fractions, and our task is to transform it into a more manageable form. Specifically, we'll be dealing with the expression $\frac{3x}{x+2} + \frac{4x}{x-2}$. Our objective is to find the values of a, b, and c that make this expression equivalent to $\frac{ax^2 + bx}{x^2 + c}$. This involves combining fractions, expanding, and then comparing the resulting expression with the target form. It's a journey of algebraic manipulation, where each step is carefully considered to arrive at the solution.

Combining the Fractions

The first step involves combining the two fractions into a single one. This means finding a common denominator and then adjusting the numerators accordingly. This process might seem a bit daunting at first, but with a systematic approach, it becomes quite manageable. Remember, the key to success here is to find the least common denominator (LCD) of the fractions. In this case, since the denominators are x + 2 and x - 2, and they don't share any common factors, the LCD is simply their product: (x + 2)(x - 2).

To combine the fractions, we need to rewrite each fraction with this common denominator. This is achieved by multiplying the numerator and denominator of each fraction by the appropriate factor. For the first fraction, we multiply both the numerator and denominator by (x - 2). For the second fraction, we multiply both the numerator and denominator by (x + 2). This ensures that we're essentially multiplying by 1 (in the form of a fraction), so we're not changing the value of the expressions, just their form. Once we've done this, the two fractions will have the same denominator, which allows us to add their numerators.

Here’s how it looks:

$\frac{3x}{x+2} + \frac{4x}{x-2} = \frac{3x(x-2)}{(x+2)(x-2)} + \frac{4x(x+2)}{(x-2)(x+2)}$

By following these steps, we make sure to keep the equation balanced while we get to our desired form. This is what we call simplifying algebraic expressions.

Expanding and Simplifying the Numerator

After we've found our common denominator and adjusted our numerators, the next phase in our algebraic adventure involves expanding the numerator and simplifying the expression. In this case, we have a mix of multiplications and additions. We need to follow the order of operations, paying close attention to the distributive property and combining like terms. This stage is where the individual terms are expanded, the like terms are grouped, and any possible simplifications are done. This process leads us closer to our goal: the target form of $\frac{ax^2 + bx}{x^2 + c}$. It's like building with blocks – each expansion and combination is a step towards completing the final structure.

Let’s start expanding the numerator of each term:

• $3x(x - 2) = 3x^2 - 6x$
• $4x(x + 2) = 4x^2 + 8x$

Now, substitute these back into the expression:

$\frac{3x^2 - 6x}{(x+2)(x-2)} + \frac{4x^2 + 8x}{(x+2)(x-2)}$

Since we now have a common denominator, we can combine the numerators:

$\frac{(3x^2 - 6x) + (4x^2 + 8x)}{(x+2)(x-2)}$

Simplifying the numerator gives us:

$\frac{7x^2 + 2x}{(x+2)(x-2)}$

The next step is to simplify the denominator.

Simplifying the Denominator

Now, let's turn our attention to the denominator. We have the expression (x + 2)(x - 2). Recognizing this pattern is key. It's an instance of the