Solve S1/x^2-2x-1/x^2+2x

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

Evaluate

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

$$x\neq 0\text{ and }|x|\neq 2$$

Short Solution Steps

Express S\times \frac{1}{x^{2}-2x} as a single fraction.

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

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

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

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

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

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

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

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

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

Expand x\left(x-2\right)\left(x+2\right).

$$\frac{Sx+2S-x+2}{x^{3}-4x}$$

Differentiate w.r.t. x

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

$$x\neq 0\text{ and }|x|\neq 2$$