Solve (x^3-2x^2/2x^2-x^2-4/4x):x-2/2+1/2

$$( \frac { x ^ { 3 } - 2 x ^ { 2 } } { 2 x ^ { 2 } } - \frac { x ^ { 2 } - 4 } { 4 x } ) : \frac { x - 2 } { 2 } + \frac { 1 } { 2 }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{3}-2x^{2}}{2x^{2}}.

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

Cancel out x^{2} in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

Cancel out 2 in both numerator and denominator.

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

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

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

Cancel out x-2 in both numerator and denominator.

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

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

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

Since \frac{x-2}{2x} and \frac{x}{2x} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-2+x.

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

Factor the expressions that are not already factored in \frac{2x-2}{2x}.

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

Cancel out 2 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{3}-2x^{2}}{2x^{2}}.

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

Cancel out x^{2} in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

Cancel out 2 in both numerator and denominator.

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

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

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

Cancel out x-2 in both numerator and denominator.

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

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

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

Since \frac{x-2}{2x} and \frac{x}{2x} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-2+x.

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

Factor the expressions that are not already factored in \frac{2x-2}{2x}.

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

Cancel out 2 in both numerator and denominator.

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