Solve x/x-2+x/x-1+x/x^2-3x+2=

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

Evaluate

$$\frac{2x}{x-2}$$

Short Solution Steps

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

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

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

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

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

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

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

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

Factor x^{2}-3x+2.

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

Since \frac{2x^{2}-3x}{\left(x-2\right)\left(x-1\right)} and \frac{x}{\left(x-2\right)\left(x-1\right)} have the same denominator, add them by adding their numerators.

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

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

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

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

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

Cancel out x-1 in both numerator and denominator.

$$\frac{2x}{x-2}$$

Factor

$$\frac{2x}{x-2}$$