Solve x^2-1/(x-1)^2divx^2+x/x-1+2/x

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

Evaluate

$$\frac{3}{x}$$

Short Solution Steps

Divide \frac{x^{2}-1}{\left(x-1\right)^{2}} by \frac{x^{2}+x}{x-1} by multiplying \frac{x^{2}-1}{\left(x-1\right)^{2}} by the reciprocal of \frac{x^{2}+x}{x-1}.

$$\frac{\left(x^{2}-1\right)\left(x-1\right)}{\left(x-1\right)^{2}\left(x^{2}+x\right)}+\frac{2}{x}$$

Cancel out x-1 in both numerator and denominator.

$$\frac{x^{2}-1}{\left(x-1\right)\left(x^{2}+x\right)}+\frac{2}{x}$$

Factor the expressions that are not already factored in \frac{x^{2}-1}{\left(x-1\right)\left(x^{2}+x\right)}.

$$\frac{\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}+\frac{2}{x}$$

Cancel out \left(x-1\right)\left(x+1\right) in both numerator and denominator.

$$\frac{1}{x}+\frac{2}{x}$$

Since \frac{1}{x} and \frac{2}{x} have the same denominator, add them by adding their numerators. Add 1 and 2 to get 3.

$$\frac{3}{x}$$

Expand

$$\frac{3}{x}$$

Short Solution Steps

Divide \frac{x^{2}-1}{\left(x-1\right)^{2}} by \frac{x^{2}+x}{x-1} by multiplying \frac{x^{2}-1}{\left(x-1\right)^{2}} by the reciprocal of \frac{x^{2}+x}{x-1}.

$$\frac{\left(x^{2}-1\right)\left(x-1\right)}{\left(x-1\right)^{2}\left(x^{2}+x\right)}+\frac{2}{x}$$

Cancel out x-1 in both numerator and denominator.

$$\frac{x^{2}-1}{\left(x-1\right)\left(x^{2}+x\right)}+\frac{2}{x}$$

Factor the expressions that are not already factored in \frac{x^{2}-1}{\left(x-1\right)\left(x^{2}+x\right)}.

$$\frac{\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}+\frac{2}{x}$$

Cancel out \left(x-1\right)\left(x+1\right) in both numerator and denominator.

$$\frac{1}{x}+\frac{2}{x}$$

Since \frac{1}{x} and \frac{2}{x} have the same denominator, add them by adding their numerators. Add 1 and 2 to get 3.

$$\frac{3}{x}$$