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

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

Evaluate

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Short Solution Steps

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

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

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

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

Combine like terms in -x+1+x+1.

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

Factor 1-x^{2}.

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

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

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

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

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

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

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

Combine like terms in -2-2x^{2}+4.

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

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

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

Extract the negative sign in -1-x.

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

Cancel out 2\left(x-1\right)\left(x+1\right) in both numerator and denominator.

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Factor

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