Problem 52 - Algebra and Trigonometry

Question

Perform the addition or subtraction and simplify. $$ \frac{5}{2 x-3}-\frac{3}{(2 x-3)^{2}} $$

Step-by-Step Solution

Answer

\( \frac{2(5x - 9)}{(2x - 3)^2} \) is the simplified result.

Step 1: Understanding the Problem

We need to perform the subtraction of two fractions: \( \frac{5}{2x-3} \) and \( \frac{3}{(2x-3)^2} \). Both fractions need a common denominator for subtraction.

Step 2: Common Denominator

The common denominator for the fractions \( \frac{5}{2x-3} \) and \( \frac{3}{(2x-3)^2} \) is \((2x-3)^2\). This is because \((2x-3)^2\) is the least common multiple of the two denominators \((2x-3)\) and \((2x-3)^2\).

Step 3: Rewrite First Fraction

To write \( \frac{5}{2x-3} \) with the common denominator \((2x-3)^2\), multiply both the numerator and the denominator by \((2x-3)\). Hence, it becomes \( \frac{5(2x-3)}{(2x-3)^2} \).

Step 4: Subtract the Fractions

Now, perform the subtraction: \[ \frac{5(2x-3)}{(2x-3)^2} - \frac{3}{(2x-3)^2} = \frac{5(2x-3) - 3}{(2x-3)^2} \].

Step 5: Simplify the Numerator

Expand the numerator: \[ 5(2x-3) = 10x - 15 \]. Simplifying the full expression gives: \( \frac{10x - 15 - 3}{(2x-3)^2} = \frac{10x - 18}{(2x-3)^2} \).

Step 6: Factor and Simplify if Possible

Factor the numerator \(10x - 18\) to \(2(5x - 9)\). Check for further simplification, but since \(5x - 9\) doesn't cancel with anything in the denominator, it remains as \( \frac{2(5x - 9)}{(2x-3)^2} \).

Key Concepts

Common Denominator

When working with algebraic fractions, a common denominator is essential for adding or subtracting. This is similar to dealing with simple numerical fractions.
A common denominator is simply a shared base for two fractions, effectively giving them the same bottom number for hassle-free arithmetic.

• In the given exercise, our fractions are \( \frac{5}{2x-3} \) and \( \frac{3}{(2x-3)^2} \).
• These have denominators of \((2x-3)\) and \((2x-3)^2\) respectively.

Here, the least common denominator turns out to be \((2x-3)^2\). This is because it is the smallest expression that both original denominators can multiply into. It ensures the same bottom part of the fraction is used for both terms without introducing new factors.
Understanding and finding the common denominator is vital to correctly perform fraction subtraction in algebra.

Subtraction of Fractions

Subtracting fractions with algebra involves managing both numerators and denominators carefully.
First, ensure both fractions have a common denominator before proceeding with subtraction.
Here's how it works for our exercise:

• First, rewrite \( \frac{5}{2x-3} \) using the common denominator, by multiplying both its numerator and denominator by the additional \((2x-3)\).
• This becomes \( \frac{5(2x-3)}{(2x-3)^2} \).
• Now, the subtraction can proceed: \[ \frac{5(2x-3)}{(2x-3)^2} - \frac{3}{(2x-3)^2} = \frac{5(2x-3) - 3}{(2x-3)^2} \].

Notice that subtraction affects only the numerators since the denominators are already the same.
Careful expansion, subtraction, and simplification of the numerators is crucial in this process.

Simplifying Expressions

Simplifying expressions involves making them as concise and straightforward as possible without changing their value.
In our problem, we must simplify the expression we obtained from performing the subtraction. This means:

• Expanding expressions like \( 5(2x-3) = 10x - 15 \).
• Subtracting: \( 10x - 15 - 3 = 10x - 18 \).

After getting \( \frac{10x - 18}{(2x-3)^2} \), further simplification involves:

• Factoring the numerator: \( 10x - 18 \) becomes \( 2(5x - 9) \).
• Checking if these factors can cancel with anything in the denominator.

Here, the denominator remains \((2x-3)^2\) and doesn't simplify further with the numerator.
Thus, the final simplified form is: \( \frac{2(5x - 9)}{(2x-3)^2} \).
Simplifying is all about recognizing patterns or redundancies, ensuring you express your answer in the simplest form possible.