Solve f(x)=(x-1/1-x):x^2-x+1/x^2-2x+1

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

Evaluate

$$x-1$$

Short Solution Steps

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

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

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

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

Do the multiplications in x\left(1-x\right)-1.

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

Divide \frac{x-x^{2}-1}{1-x} by \frac{x^{2}-x+1}{x^{2}-2x+1} by multiplying \frac{x-x^{2}-1}{1-x} by the reciprocal of \frac{x^{2}-x+1}{x^{2}-2x+1}.

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

Extract the negative sign in x-x^{2}-1.

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

Cancel out x^{2}-x+1 in both numerator and denominator.

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

To find the opposite of x^{2}-2x+1, find the opposite of each term.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -1+x.

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

Cancel out -x+1 in both numerator and denominator.

$$-\left(-x+1\right)$$

Expand the expression.

$$x-1$$

Expand

$$x-1$$

Short Solution Steps

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

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

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

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

Do the multiplications in x\left(1-x\right)-1.

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

Divide \frac{x-x^{2}-1}{1-x} by \frac{x^{2}-x+1}{x^{2}-2x+1} by multiplying \frac{x-x^{2}-1}{1-x} by the reciprocal of \frac{x^{2}-x+1}{x^{2}-2x+1}.

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

Extract the negative sign in x-x^{2}-1.

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

Cancel out x^{2}-x+1 in both numerator and denominator.

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

To find the opposite of x^{2}-2x+1, find the opposite of each term.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -1+x.

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

Cancel out -x+1 in both numerator and denominator.

$$-\left(-x+1\right)$$

Expand the expression.

$$x-1$$