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

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

Evaluate

$$\frac{1}{2x}$$

Short Solution Steps

Divide \frac{x-2}{x^{2}+2x} by \frac{x^{2}-4x+4}{x^{2}-4} by multiplying \frac{x-2}{x^{2}+2x} by the reciprocal of \frac{x^{2}-4x+4}{x^{2}-4}.

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

Factor the expressions that are not already factored in \frac{\left(x-2\right)\left(x^{2}-4\right)}{\left(x^{2}+2x\right)\left(x^{2}-4x+4\right)}.

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

Cancel out \left(x+2\right)\left(x-2\right)^{2} in both numerator and denominator.

$$\frac{1}{x}-\frac{1}{2x}$$

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

$$\frac{2}{2x}-\frac{1}{2x}$$

Since \frac{2}{2x} and \frac{1}{2x} have the same denominator, subtract them by subtracting their numerators. Subtract 1 from 2 to get 1.

$$\frac{1}{2x}$$

Expand

$$\frac{1}{2x}$$

Short Solution Steps

Divide \frac{x-2}{x^{2}+2x} by \frac{x^{2}-4x+4}{x^{2}-4} by multiplying \frac{x-2}{x^{2}+2x} by the reciprocal of \frac{x^{2}-4x+4}{x^{2}-4}.

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

Factor the expressions that are not already factored in \frac{\left(x-2\right)\left(x^{2}-4\right)}{\left(x^{2}+2x\right)\left(x^{2}-4x+4\right)}.

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

Cancel out \left(x+2\right)\left(x-2\right)^{2} in both numerator and denominator.

$$\frac{1}{x}-\frac{1}{2x}$$

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

$$\frac{2}{2x}-\frac{1}{2x}$$

Since \frac{2}{2x} and \frac{1}{2x} have the same denominator, subtract them by subtracting their numerators. Subtract 1 from 2 to get 1.

$$\frac{1}{2x}$$