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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Combine like terms in x-2x+4.

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

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

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

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

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

Do the multiplications in -x+4+x\left(x-2\right).

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

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

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

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

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

Differentiate w.r.t. x

$$\frac{8-12x+4x^{2}-x^{3}}{x^{2}\left(x-2\right)^{3}}$$