Question

Simplify each complex fraction. ﻿6x2−25+x1x−5\frac{\frac{6}{$$x^2 - 25$$} + x}{\frac{1}{x - 5}}
x−51​x2−256​+x​

Accepted Answer

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

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

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Key Concepts

Complex Fractions

Factoring Quadratic Expressions

Operations with Rational Expressions

Simplify each complex fraction. 6x2−25+x1x−5\(\frac{\frac{6}{x^2 - 25}\) + x}{\(\frac{1}{x - 5}\)}x−51x2−256+x