Solve 3x/x^2-1-1/x(x-1)-2x^2+1/x(x^2-1)

$$\frac { 3 x } { x ^ { 2 } - 1 } - \frac { 1 } { x ( x - 1 ) } - \frac { 2 x ^ { 2 } + 1 } { x ( x ^ { 2 } - 1 ) }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-1.

$$\frac{3x}{\left(x-1\right)\left(x+1\right)}-\frac{1}{x\left(x-1\right)}-\frac{2x^{2}+1}{x\left(x^{2}-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x\left(x-1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{3x}{\left(x-1\right)\left(x+1\right)} times \frac{x}{x}. Multiply \frac{1}{x\left(x-1\right)} times \frac{x+1}{x+1}.

$$\frac{3xx}{x\left(x-1\right)\left(x+1\right)}-\frac{x+1}{x\left(x-1\right)\left(x+1\right)}-\frac{2x^{2}+1}{x\left(x^{2}-1\right)}$$

Since \frac{3xx}{x\left(x-1\right)\left(x+1\right)} and \frac{x+1}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3xx-\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}-\frac{2x^{2}+1}{x\left(x^{2}-1\right)}$$

Do the multiplications in 3xx-\left(x+1\right).

$$\frac{3x^{2}-x-1}{x\left(x-1\right)\left(x+1\right)}-\frac{2x^{2}+1}{x\left(x^{2}-1\right)}$$

Factor x\left(x^{2}-1\right).

$$\frac{3x^{2}-x-1}{x\left(x-1\right)\left(x+1\right)}-\frac{2x^{2}+1}{x\left(x-1\right)\left(x+1\right)}$$

Since \frac{3x^{2}-x-1}{x\left(x-1\right)\left(x+1\right)} and \frac{2x^{2}+1}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3x^{2}-x-1-\left(2x^{2}+1\right)}{x\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in 3x^{2}-x-1-\left(2x^{2}+1\right).

$$\frac{3x^{2}-x-1-2x^{2}-1}{x\left(x-1\right)\left(x+1\right)}$$

Combine like terms in 3x^{2}-x-1-2x^{2}-1.

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

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

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

Cancel out x+1 in both numerator and denominator.

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

Expand x\left(x-1\right).

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

Expand

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

Short Solution Steps

Factor x^{2}-1.

$$\frac{3x}{\left(x-1\right)\left(x+1\right)}-\frac{1}{x\left(x-1\right)}-\frac{2x^{2}+1}{x\left(x^{2}-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x\left(x-1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{3x}{\left(x-1\right)\left(x+1\right)} times \frac{x}{x}. Multiply \frac{1}{x\left(x-1\right)} times \frac{x+1}{x+1}.

$$\frac{3xx}{x\left(x-1\right)\left(x+1\right)}-\frac{x+1}{x\left(x-1\right)\left(x+1\right)}-\frac{2x^{2}+1}{x\left(x^{2}-1\right)}$$

Since \frac{3xx}{x\left(x-1\right)\left(x+1\right)} and \frac{x+1}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3xx-\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}-\frac{2x^{2}+1}{x\left(x^{2}-1\right)}$$

Do the multiplications in 3xx-\left(x+1\right).

$$\frac{3x^{2}-x-1}{x\left(x-1\right)\left(x+1\right)}-\frac{2x^{2}+1}{x\left(x^{2}-1\right)}$$

Factor x\left(x^{2}-1\right).

$$\frac{3x^{2}-x-1}{x\left(x-1\right)\left(x+1\right)}-\frac{2x^{2}+1}{x\left(x-1\right)\left(x+1\right)}$$

Since \frac{3x^{2}-x-1}{x\left(x-1\right)\left(x+1\right)} and \frac{2x^{2}+1}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3x^{2}-x-1-\left(2x^{2}+1\right)}{x\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in 3x^{2}-x-1-\left(2x^{2}+1\right).

$$\frac{3x^{2}-x-1-2x^{2}-1}{x\left(x-1\right)\left(x+1\right)}$$

Combine like terms in 3x^{2}-x-1-2x^{2}-1.

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

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

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

Cancel out x+1 in both numerator and denominator.

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

Expand x\left(x-1\right).

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