Subtract the following rational expressions and write the difference in simplest form if possible.
$\frac{x^2}{x^2-25}-\frac{5x}{25-x^2}$

• A. $\frac{x}{x+5}$
• B. $\frac{x}{x-5}$
• C. $\frac{x}{(x-5)(x+5)}$
• D. $\frac{x+5}{x-5}$

Answer: (B)

Verified step by step guidance

1. Identify the given rational expressions: \(\frac{x^2}{x^2 - 25} - \frac{5x}{25 - x^2}\).
2. Notice that the denominators \(x^2 - 25\) and \$25 - x^2\( are related. Recall that \)x^2 - 25\( factors as \)(x - 5)(x + 5)\(, and \)25 - x^2\( is the negative of \)x^2 - 25\(, so \)25 - x^2 = -(x^2 - 25) = -(x - 5)(x + 5)$.
3. Rewrite the second fraction to have the same denominator as the first by factoring out the negative sign: \(\frac{5x}{25 - x^2} = \frac{5x}{-(x - 5)(x + 5)} = -\frac{5x}{(x - 5)(x + 5)}\).
4. Now express both fractions with the common denominator \((x - 5)(x + 5)\): the first fraction is \(\frac{x^2}{(x - 5)(x + 5)}\), and the second is \(-\frac{5x}{(x - 5)(x + 5)}\). The subtraction becomes \(\frac{x^2}{(x - 5)(x + 5)} - \left(-\frac{5x}{(x - 5)(x + 5)}\right)\).
5. Combine the numerators over the common denominator: \(\frac{x^2 + 5x}{(x - 5)(x + 5)}\). Then factor the numerator if possible and simplify the expression by canceling common factors.