Problem 19 - Algebra 1

Question

Simplify the expression. $$\frac{4 x}{x^{2}-9} \cdot \frac{x-3}{8 x^{2}+12 x}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{1}{x+3}\).

Step 1: Factor the denominators

Factorize the denominators by finding common factors or difference of squares wherever possible. So, \(x^{2}-9\) is the difference of squares which will factor into \((x+3)(x-3)\) and simplify \(8 x^{2}+12 x\) by taking out the common factor \(4x\), which will result into \(4x(2x+3)\). Hence the expression becomes, \(\frac{4 x}{(x+3)(x-3)} \cdot \frac{x-3}{4x(2x+3)}\).

Step 2: Cancel Similar Terms

Cancelling out the similar terms from the numerator and the denominator will simplify the expression further. Here, \(4x\), \(x-3\), and \(2x+3\) can be cancelled. After cancellation, the expression is, \(\frac{1}{x+3}\).

Step 3: Simplified Expression

So, the simplified form of the given expression is \(\frac{1}{x+3}\).

Key Concepts

Difference of Squares

When simplifying algebraic expressions, recognizing patterns like the "difference of squares" is key. The difference of squares formula states that if you have two square terms separated by a subtraction sign, such as

• \(a^2 - b^2\),

you can factor this into

• \((a-b)(a+b)\).

This formula is incredibly useful for breaking down and simplifying expressions.
In our exercise, the term \(x^2 - 9\) is an example.
The expression \(9\) is the same as \(3^2\), so it fits the difference of squares form:\(x^2 - 3^2\).
Thus, it can be factored to

• \((x + 3)(x - 3)\).

This step makes simplifying the rest of the expression easier. Always look for these patterns to help break down complex expressions.

Factoring

Factoring is a fundamental concept in algebra that involves breaking down complex expressions into simpler ones or identifying common factors in terms. This process is pivotal in simplifying expressions.
For example, in the current exercise, besides the difference of squares, we dealt with the term \(8x^2 + 12x\).
Here, what you need to do is factor out the greatest common factor (GCF).
Identify the largest factor that both terms can share.

• In this case, it is \(4x\).

Factoring \(4x\) out from both terms of \(8x^2 + 12x\) results in

• \(4x(2x + 3)\).

Understanding how to factor each component will make simplifying significantly more intuitive.
In more complex expressions, the skill of factoring makes them approachable and less daunting.

Simplifying Rational Expressions

Rational expressions are quotients of polynomial expressions, and simplifying them is very similar to simplifying fractions.
The goal is to cancel out common factors in the numerator and the denominator.
In our example, after factoring, the expression was transformed into

• \(\frac{4x}{(x+3)(x-3)} \cdot \frac{x-3}{4x(2x+3)}\).

Notice how certain terms appear both in the numerator and denominator such as \(4x\), \(x-3\), and \(2x+3\).
Cancelling these matching forms simplifies the entire expression to

• \(\frac{1}{x+3}\).

The ability to identify and cancel common terms is a powerful tool.
Always ensure that the denominator never becomes zero, as this would make the expression undefined.
In algebraic contexts, simplifying means making the expression as straightforward as possible, while maintaining its equivalence.