Solve 2x^2-2x+1/x^2-x+x/1-x/1+frac{2{x}-3/x^2}

$$\frac{ \frac{ 2 { x }^{ 2 } -2x+1 }{ { x }^{ 2 } -x } + \frac{ x }{ 1-x } }{ 1+ \frac{ 2 }{ x } - \frac{ 3 }{ { x }^{ 2 } } }$$

Evaluate

$$\frac{x}{x+3}$$

Short Solution Steps

Factor x^{2}-x.

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

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

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

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

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

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

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

Combine like terms in 2x^{2}-2x+1-x^{2}.

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

Factor the expressions that are not already factored in \frac{x^{2}-2x+1}{x\left(x-1\right)}.

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

Cancel out x-1 in both numerator and denominator.

$$\frac{\frac{x-1}{x}}{1+\frac{2}{x}-\frac{3}{x^{2}}}$$

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

$$\frac{\frac{x-1}{x}}{\frac{x}{x}+\frac{2}{x}-\frac{3}{x^{2}}}$$

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

$$\frac{\frac{x-1}{x}}{\frac{x+2}{x}-\frac{3}{x^{2}}}$$

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

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

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

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

Do the multiplications in \left(x+2\right)x-3.

$$\frac{\frac{x-1}{x}}{\frac{x^{2}+2x-3}{x^{2}}}$$

Divide \frac{x-1}{x} by \frac{x^{2}+2x-3}{x^{2}} by multiplying \frac{x-1}{x} by the reciprocal of \frac{x^{2}+2x-3}{x^{2}}.

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

Cancel out x in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

$$\frac{x}{x+3}$$

Expand

$$\frac{x}{x+3}$$

Short Solution Steps

Factor x^{2}-x.

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

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

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

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

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

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

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

Combine like terms in 2x^{2}-2x+1-x^{2}.

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

Factor the expressions that are not already factored in \frac{x^{2}-2x+1}{x\left(x-1\right)}.

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

Cancel out x-1 in both numerator and denominator.

$$\frac{\frac{x-1}{x}}{1+\frac{2}{x}-\frac{3}{x^{2}}}$$

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

$$\frac{\frac{x-1}{x}}{\frac{x}{x}+\frac{2}{x}-\frac{3}{x^{2}}}$$

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

$$\frac{\frac{x-1}{x}}{\frac{x+2}{x}-\frac{3}{x^{2}}}$$

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

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

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

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

Do the multiplications in \left(x+2\right)x-3.

$$\frac{\frac{x-1}{x}}{\frac{x^{2}+2x-3}{x^{2}}}$$

Divide \frac{x-1}{x} by \frac{x^{2}+2x-3}{x^{2}} by multiplying \frac{x-1}{x} by the reciprocal of \frac{x^{2}+2x-3}{x^{2}}.

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

Cancel out x in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

$$\frac{x}{x+3}$$