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

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

Evaluate

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

Short Solution Steps

Factor x^{3}-4x. Factor x^{3}-4x^{2}+4x.

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

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\right)^{2} is x\left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{x+3}{x\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{x-3}{x\left(x-2\right)^{2}} times \frac{x+2}{x+2}.

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

Since \frac{\left(x+3\right)\left(x-2\right)}{x\left(x+2\right)\left(x-2\right)^{2}} and \frac{\left(x-3\right)\left(x+2\right)}{x\left(x+2\right)\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+3\right)\left(x-2\right)-\left(x-3\right)\left(x+2\right).

$$\frac{x^{2}-2x+3x-6-x^{2}-2x+3x+6}{x\left(x+2\right)\left(x-2\right)^{2}}$$

Combine like terms in x^{2}-2x+3x-6-x^{2}-2x+3x+6.

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

Cancel out x in both numerator and denominator.

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

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

$$\frac{2}{x^{3}-2x^{2}-4x+8}$$

Expand

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

Short Solution Steps

Factor x^{3}-4x. Factor x^{3}-4x^{2}+4x.

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

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\right)^{2} is x\left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{x+3}{x\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{x-3}{x\left(x-2\right)^{2}} times \frac{x+2}{x+2}.

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

Since \frac{\left(x+3\right)\left(x-2\right)}{x\left(x+2\right)\left(x-2\right)^{2}} and \frac{\left(x-3\right)\left(x+2\right)}{x\left(x+2\right)\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+3\right)\left(x-2\right)-\left(x-3\right)\left(x+2\right).

$$\frac{x^{2}-2x+3x-6-x^{2}-2x+3x+6}{x\left(x+2\right)\left(x-2\right)^{2}}$$

Combine like terms in x^{2}-2x+3x-6-x^{2}-2x+3x+6.

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

Cancel out x in both numerator and denominator.

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

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

$$\frac{2}{x^{3}-2x^{2}-4x+8}$$