Question

simplify the rational expression below
\\(\frac{3x - 5}{x^2 - 25} - \frac{2}{x + 5}\\)

Explanation:

Step1: Factor the denominator

First, factor the denominator \(x^2 - 25\) using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\). So, \(x^2 - 25=(x + 5)(x - 5)\). The expression becomes \(\frac{3x-5}{(x + 5)(x - 5)}-\frac{2}{x + 5}\).

Step2: Find a common denominator

The common denominator for the two fractions is \((x + 5)(x - 5)\). Rewrite the second fraction with the common denominator: \(\frac{2}{x + 5}=\frac{2(x - 5)}{(x + 5)(x - 5)}\). Now the expression is \(\frac{3x-5}{(x + 5)(x - 5)}-\frac{2(x - 5)}{(x + 5)(x - 5)}\).

Step3: Subtract the numerators

Subtract the numerators while keeping the common denominator: \(\frac{(3x - 5)-2(x - 5)}{(x + 5)(x - 5)}\). Expand the numerator: \(3x - 5-2x + 10\). Combine like terms: \((3x-2x)+(-5 + 10)=x + 5\). So the expression becomes \(\frac{x + 5}{(x + 5)(x - 5)}\).

Step4: Cancel common factors

Cancel out the common factor \((x + 5)\) from the numerator and the denominator (assuming \(x
eq - 5\) and \(x
eq5\)): \(\frac{1}{x - 5}\).

Answer:

\(\frac{1}{x - 5}\)