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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in \left(x+2\right)\left(x-2\right)-\left(x-1\right)x.

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in \left(x+2\right)\left(x-2\right)-\left(x-1\right)x.

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

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

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

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

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