Solve x/x+2-8/x^2-4+2/x-2

$$\frac{ x }{ x+2 } - \frac{ 8 }{ { x }^{ 2 } -4 } + \frac{ 2 }{ x-2 }$$

Evaluate

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Short Solution Steps

Factor x^{2}-4.

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

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

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

Since \frac{x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{8}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(x-2\right)-8.

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

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

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

Cancel out x+2 in both numerator and denominator.

$$\frac{x-4}{x-2}+\frac{2}{x-2}$$

Since \frac{x-4}{x-2} and \frac{2}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{x-4+2}{x-2}$$

Combine like terms in x-4+2.

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

Cancel out x-2 in both numerator and denominator.

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Factor

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