What is (x/(x+1)-1/x)/(x/(x+1)+1/x) ?

The solution to (x/(x+1)-1/x)/(x/(x+1)+1/x) is (x^2-x-1)/(x^2+x+1)

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simplify (x/(x+1)-1/x)/(x/(x+1)+1/x)

simplify

\frac{\frac{x}{x + 1} - \frac{1}{x}}{\frac{x}{x + 1} + \frac{1}{x}}

\frac{x ^{2} - x - 1}{x ^{2} + x + 1}

Solution steps

\frac{\frac{x}{x + 1} - \frac{1}{x}}{\frac{x}{x + 1} + \frac{1}{x}}

Join

\frac{x}{x + 1} - \frac{1}{x} : \frac{x ^{2} - x - 1}{x ( x + 1 )}

= \frac{\frac{x ^{2} - x - 1}{x ( x + 1 )}}{\frac{x}{x + 1} + \frac{1}{x}}

Join

\frac{x}{x + 1} + \frac{1}{x} : \frac{x ^{2} + x + 1}{x ( x + 1 )}

= \frac{\frac{x ^{2} - x - 1}{x ( x + 1 )}}{\frac{x ^{2} + x + 1}{x ( x + 1 )}}

Apply the fraction rule:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( x ^{2} - x - 1 ) x ( x + 1 )}{x ( x + 1 ) ( x ^{2} + x + 1 )}

Cancel

\frac{( x ^{2} - x - 1 ) x ( x + 1 )}{x ( x + 1 ) ( x ^{2} + x + 1 )} : \frac{x ^{2} - x - 1}{x ^{2} + x + 1}

= \frac{x ^{2} - x - 1}{x ^{2} + x + 1}

\frac{2 x ^{2} - x - 1}{x - 1}

\frac{sin ^{2} ( y )}{1 - cos ( y )}

\frac{2 a}{2}

(1 + i)^{18}

(5 + 2 i) (3 - 2 i)

What is (x/(x+1)-1/x)/(x/(x+1)+1/x) ?
The solution to (x/(x+1)-1/x)/(x/(x+1)+1/x) is (x^2-x-1)/(x^2+x+1)