Solve (1-1/x-1)divx^2-4/x^2-2x+1

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-1}{x-1}.

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

Since \frac{x-1}{x-1} and \frac{1}{x-1} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in x-1-1.

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

Express \frac{\frac{x-2}{x-1}}{x^{2}} as a single fraction.

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

Factor x^{2}-2x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)x^{2} and \left(x-1\right)^{2} is x^{2}\left(x-1\right)^{2}. Multiply \frac{x-2}{\left(x-1\right)x^{2}} times \frac{x-1}{x-1}. Multiply \frac{4}{\left(x-1\right)^{2}} times \frac{x^{2}}{x^{2}}.

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

Since \frac{\left(x-2\right)\left(x-1\right)}{x^{2}\left(x-1\right)^{2}} and \frac{4x^{2}}{x^{2}\left(x-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2\right)\left(x-1\right)-4x^{2}.

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

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

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

Expand x^{2}\left(x-1\right)^{2}.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-1}{x-1}.

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

Since \frac{x-1}{x-1} and \frac{1}{x-1} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in x-1-1.

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

Express \frac{\frac{x-2}{x-1}}{x^{2}} as a single fraction.

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

Factor x^{2}-2x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)x^{2} and \left(x-1\right)^{2} is x^{2}\left(x-1\right)^{2}. Multiply \frac{x-2}{\left(x-1\right)x^{2}} times \frac{x-1}{x-1}. Multiply \frac{4}{\left(x-1\right)^{2}} times \frac{x^{2}}{x^{2}}.

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

Since \frac{\left(x-2\right)\left(x-1\right)}{x^{2}\left(x-1\right)^{2}} and \frac{4x^{2}}{x^{2}\left(x-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2\right)\left(x-1\right)-4x^{2}.

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

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

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

Expand x^{2}\left(x-1\right)^{2}.

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