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

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

Evaluate

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

Short Solution Steps

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

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

Since -\frac{1}{x^{2}} and \frac{x}{x^{2}} have the same denominator, add them by adding their numerators.

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

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

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

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

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

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

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

Combine like terms in -x^{2}+4x-4+x^{3}-4x^{2}+4x+x^{2}.

$$\frac{-4x^{2}+8x-4+x^{3}}{x^{2}\left(x-2\right)^{2}}-\frac{1}{2-x}$$

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

$$\frac{-4x^{2}+8x-4+x^{3}}{x^{2}\left(x-2\right)^{2}}-\frac{-\left(x-2\right)x^{2}}{x^{2}\left(x-2\right)^{2}}$$

Since \frac{-4x^{2}+8x-4+x^{3}}{x^{2}\left(x-2\right)^{2}} and \frac{-\left(x-2\right)x^{2}}{x^{2}\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x^{2}+8x-4+x^{3}-\left(-\left(x-2\right)x^{2}\right)}{x^{2}\left(x-2\right)^{2}}$$

Do the multiplications in -4x^{2}+8x-4+x^{3}-\left(-\left(x-2\right)x^{2}\right).

$$\frac{-4x^{2}+8x-4+x^{3}+x^{3}-2x^{2}}{x^{2}\left(x-2\right)^{2}}$$

Combine like terms in -4x^{2}+8x-4+x^{3}+x^{3}-2x^{2}.

$$\frac{-6x^{2}+8x-4+2x^{3}}{x^{2}\left(x-2\right)^{2}}$$

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

$$\frac{-6x^{2}+8x-4+2x^{3}}{x^{4}-4x^{3}+4x^{2}}$$

Expand

$$\frac{2\left(x^{3}-3x^{2}+4x-2\right)}{\left(x\left(x-2\right)\right)^{2}}$$

Short Solution Steps

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

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

Since -\frac{1}{x^{2}} and \frac{x}{x^{2}} have the same denominator, add them by adding their numerators.

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

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

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

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

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

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

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

Combine like terms in -x^{2}+4x-4+x^{3}-4x^{2}+4x+x^{2}.

$$\frac{-4x^{2}+8x-4+x^{3}}{x^{2}\left(x-2\right)^{2}}-\frac{1}{2-x}$$

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

$$\frac{-4x^{2}+8x-4+x^{3}}{x^{2}\left(x-2\right)^{2}}-\frac{-\left(x-2\right)x^{2}}{x^{2}\left(x-2\right)^{2}}$$

Since \frac{-4x^{2}+8x-4+x^{3}}{x^{2}\left(x-2\right)^{2}} and \frac{-\left(x-2\right)x^{2}}{x^{2}\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x^{2}+8x-4+x^{3}-\left(-\left(x-2\right)x^{2}\right)}{x^{2}\left(x-2\right)^{2}}$$

Do the multiplications in -4x^{2}+8x-4+x^{3}-\left(-\left(x-2\right)x^{2}\right).

$$\frac{-4x^{2}+8x-4+x^{3}+x^{3}-2x^{2}}{x^{2}\left(x-2\right)^{2}}$$

Combine like terms in -4x^{2}+8x-4+x^{3}+x^{3}-2x^{2}.

$$\frac{-6x^{2}+8x-4+2x^{3}}{x^{2}\left(x-2\right)^{2}}$$

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

$$\frac{-6x^{2}+8x-4+2x^{3}}{x^{4}-4x^{3}+4x^{2}}$$