Solve x^2-5x+6/x+x-2/x^2-3x

$$\frac { x ^ { 2 } - 5 x + 6 } { x } + \frac { x - 2 } { x ^ { 2 } - 3 x }$$

Evaluate

$$\frac{\left(x-2\right)\left(x^{2}-6x+10\right)}{x\left(x-3\right)}$$

Short Solution Steps

Factor x^{2}-3x.

$$\frac{x^{2}-5x+6}{x}+\frac{x-2}{x\left(x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x\left(x-3\right) is x\left(x-3\right). Multiply \frac{x^{2}-5x+6}{x} times \frac{x-3}{x-3}.

$$\frac{\left(x^{2}-5x+6\right)\left(x-3\right)}{x\left(x-3\right)}+\frac{x-2}{x\left(x-3\right)}$$

Since \frac{\left(x^{2}-5x+6\right)\left(x-3\right)}{x\left(x-3\right)} and \frac{x-2}{x\left(x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}-5x+6\right)\left(x-3\right)+x-2}{x\left(x-3\right)}$$

Do the multiplications in \left(x^{2}-5x+6\right)\left(x-3\right)+x-2.

$$\frac{x^{3}-3x^{2}-5x^{2}+15x+6x-18+x-2}{x\left(x-3\right)}$$

Combine like terms in x^{3}-3x^{2}-5x^{2}+15x+6x-18+x-2.

$$\frac{x^{3}-8x^{2}+22x-20}{x\left(x-3\right)}$$

Expand x\left(x-3\right).

$$\frac{x^{3}-8x^{2}+22x-20}{x^{2}-3x}$$

Expand

$$\frac{x^{3}-8x^{2}+22x-20}{x\left(x-3\right)}$$

Short Solution Steps

Factor x^{2}-3x.

$$\frac{x^{2}-5x+6}{x}+\frac{x-2}{x\left(x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x\left(x-3\right) is x\left(x-3\right). Multiply \frac{x^{2}-5x+6}{x} times \frac{x-3}{x-3}.

$$\frac{\left(x^{2}-5x+6\right)\left(x-3\right)}{x\left(x-3\right)}+\frac{x-2}{x\left(x-3\right)}$$

Since \frac{\left(x^{2}-5x+6\right)\left(x-3\right)}{x\left(x-3\right)} and \frac{x-2}{x\left(x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}-5x+6\right)\left(x-3\right)+x-2}{x\left(x-3\right)}$$

Do the multiplications in \left(x^{2}-5x+6\right)\left(x-3\right)+x-2.

$$\frac{x^{3}-3x^{2}-5x^{2}+15x+6x-18+x-2}{x\left(x-3\right)}$$

Combine like terms in x^{3}-3x^{2}-5x^{2}+15x+6x-18+x-2.

$$\frac{x^{3}-8x^{2}+22x-20}{x\left(x-3\right)}$$

Expand x\left(x-3\right).

$$\frac{x^{3}-8x^{2}+22x-20}{x^{2}-3x}$$