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

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x^{2} is \left(x-1\right)x^{2}. Multiply \frac{2}{x-1} times \frac{x^{2}}{x^{2}}. Multiply \frac{x+1}{x^{2}} times \frac{x-1}{x-1}.

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

Since \frac{2x^{2}}{\left(x-1\right)x^{2}} and \frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2x^{2}-\left(x+1\right)\left(x-1\right).

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

Combine like terms in 2x^{2}-x^{2}+x-x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)x^{2} and x\left(x-1\right) is \left(x-1\right)x^{2}. Multiply \frac{2}{x\left(x-1\right)} times \frac{x}{x}.

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

Since \frac{x^{2}+1}{\left(x-1\right)x^{2}} and \frac{2x}{\left(x-1\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Cancel out x-1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x^{2} is \left(x-1\right)x^{2}. Multiply \frac{2}{x-1} times \frac{x^{2}}{x^{2}}. Multiply \frac{x+1}{x^{2}} times \frac{x-1}{x-1}.

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

Since \frac{2x^{2}}{\left(x-1\right)x^{2}} and \frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2x^{2}-\left(x+1\right)\left(x-1\right).

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

Combine like terms in 2x^{2}-x^{2}+x-x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)x^{2} and x\left(x-1\right) is \left(x-1\right)x^{2}. Multiply \frac{2}{x\left(x-1\right)} times \frac{x}{x}.

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

Since \frac{x^{2}+1}{\left(x-1\right)x^{2}} and \frac{2x}{\left(x-1\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Cancel out x-1 in both numerator and denominator.

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