Solve 1/x^2-1-1/2x^2-6x+4+3/2x^2-2x-4

$$\frac { 1 } { x ^ { 2 } - 1 } - \frac { 1 } { 2 x ^ { 2 } - 6 x + 4 } + \frac { 3 } { 2 x ^ { 2 } - 2 x - 4 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-1. Factor 2x^{2}-6x+4.

$$\frac{1}{\left(x-1\right)\left(x+1\right)}-\frac{1}{2\left(x-2\right)\left(x-1\right)}+\frac{3}{2x^{2}-2x-4}$$

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 2\left(x-2\right)\left(x-1\right) is 2\left(x-2\right)\left(x-1\right)\left(x+1\right). Multiply \frac{1}{\left(x-1\right)\left(x+1\right)} times \frac{2\left(x-2\right)}{2\left(x-2\right)}. Multiply \frac{1}{2\left(x-2\right)\left(x-1\right)} times \frac{x+1}{x+1}.

$$\frac{2\left(x-2\right)}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}-\frac{x+1}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{3}{2x^{2}-2x-4}$$

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

$$\frac{2\left(x-2\right)-\left(x+1\right)}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{3}{2x^{2}-2x-4}$$

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

$$\frac{2x-4-x-1}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{3}{2x^{2}-2x-4}$$

Combine like terms in 2x-4-x-1.

$$\frac{x-5}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{3}{2x^{2}-2x-4}$$

Factor 2x^{2}-2x-4.

$$\frac{x-5}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{3}{2\left(x-2\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-2\right)\left(x-1\right)\left(x+1\right) and 2\left(x-2\right)\left(x+1\right) is 2\left(x-2\right)\left(x-1\right)\left(x+1\right). Multiply \frac{3}{2\left(x-2\right)\left(x+1\right)} times \frac{x-1}{x-1}.

$$\frac{x-5}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{3\left(x-1\right)}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}$$

Since \frac{x-5}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)} and \frac{3\left(x-1\right)}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{x-5+3\left(x-1\right)}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in x-5+3\left(x-1\right).

$$\frac{x-5+3x-3}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}$$

Combine like terms in x-5+3x-3.

$$\frac{4x-8}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in \frac{4x-8}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}.

$$\frac{4\left(x-2\right)}{2\left(x-2\right)\left(x-1\right)\left(x+1\right)}$$

Cancel out 2\left(x-2\right) in both numerator and denominator.

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

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

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

Factor

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