Simplify each complex fraction. $\(\frac{\frac{6}{x^2 - 25}\) + x}{\(\frac{1}{x - 5}\)}$

Verified step by step guidance

1. Identify the complex fraction: the numerator is \( \frac{6}{x^{2} - 25} + x \) and the denominator is \( \frac{1}{x - 5} \).
2. Factor the quadratic expression in the denominator of the numerator: \( x^{2} - 25 \) is a difference of squares, so \( x^{2} - 25 = (x - 5)(x + 5) \). Rewrite the numerator as \( \frac{6}{(x - 5)(x + 5)} + x \).
3. To combine the terms in the numerator, express \( x \) as a fraction with denominator \( (x - 5)(x + 5) \), so rewrite \( x = \frac{x (x - 5)(x + 5)}{(x - 5)(x + 5)} \).
4. Add the two fractions in the numerator: \( \frac{6}{(x - 5)(x + 5)} + \frac{x (x - 5)(x + 5)}{(x - 5)(x + 5)} = \frac{6 + x (x - 5)(x + 5)}{(x - 5)(x + 5)} \).
5. Rewrite the entire complex fraction as a division: \( \frac{\frac{6 + x (x - 5)(x + 5)}{(x - 5)(x + 5)}}{\frac{1}{x - 5}} = \frac{6 + x (x - 5)(x + 5)}{(x - 5)(x + 5)} \times \frac{x - 5}{1} \). Then simplify by canceling common factors and simplifying the numerator expression.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression as a single fraction by finding common denominators or multiplying by the reciprocal of the denominator.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of simpler binomials. For example, x^2 - 25 is a difference of squares and factors into (x - 5)(x + 5), which helps simplify expressions and cancel common factors.

Operations with Rational Expressions

Rational expressions are fractions with polynomials in numerator and denominator. Adding, subtracting, multiplying, or dividing them requires finding common denominators, factoring, and simplifying by canceling common factors to reduce the expression.