Solve 1/x(x+2)(x-2)-2/x(x^2+4)

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

Evaluate

$$\frac{12-x^{2}}{x\left(x^{4}-16\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right)\left(x-2\right) and x\left(x^{2}+4\right) is x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right). Multiply \frac{1}{x\left(x+2\right)\left(x-2\right)} times \frac{x^{2}+4}{x^{2}+4}. Multiply \frac{2}{x\left(x^{2}+4\right)} times \frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}.

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

Since \frac{x^{2}+4}{x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right)} and \frac{2\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}+4-2\left(x-2\right)\left(x+2\right).

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

Combine like terms in x^{2}+4-2x^{2}-4x+4x+8.

$$\frac{-x^{2}+12}{x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right)}$$

Expand x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right).

$$\frac{-x^{2}+12}{x^{5}-16x}$$

Expand

$$\frac{12-x^{2}}{x\left(x^{4}-16\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right)\left(x-2\right) and x\left(x^{2}+4\right) is x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right). Multiply \frac{1}{x\left(x+2\right)\left(x-2\right)} times \frac{x^{2}+4}{x^{2}+4}. Multiply \frac{2}{x\left(x^{2}+4\right)} times \frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}.

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

Since \frac{x^{2}+4}{x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right)} and \frac{2\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}+4-2\left(x-2\right)\left(x+2\right).

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

Combine like terms in x^{2}+4-2x^{2}-4x+4x+8.

$$\frac{-x^{2}+12}{x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right)}$$

Expand x\left(x-2\right)\left(x+2\right)\left(x^{2}+4\right).

$$\frac{-x^{2}+12}{x^{5}-16x}$$