Solve x^2-5x+6/x^3-6x^2+12x-8*(4-x/x-3+2)

$$\frac { x ^ { 2 } - 5 x + 6 } { x ^ { 3 } - 6 x ^ { 2 } + 12 x - 8 } \cdot ( \frac { 4 - x } { x - 3 } + 2 )$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-5x+6}{x^{3}-6x^{2}+12x-8}.

$$\frac{\left(x-3\right)\left(x-2\right)}{\left(x-2\right)^{3}}\left(\frac{4-x}{x-3}+2\right)$$

Cancel out x-2 in both numerator and denominator.

$$\frac{x-3}{\left(x-2\right)^{2}}\left(\frac{4-x}{x-3}+2\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x-3}{x-3}.

$$\frac{x-3}{\left(x-2\right)^{2}}\left(\frac{4-x}{x-3}+\frac{2\left(x-3\right)}{x-3}\right)$$

Since \frac{4-x}{x-3} and \frac{2\left(x-3\right)}{x-3} have the same denominator, add them by adding their numerators.

$$\frac{x-3}{\left(x-2\right)^{2}}\times \frac{4-x+2\left(x-3\right)}{x-3}$$

Do the multiplications in 4-x+2\left(x-3\right).

$$\frac{x-3}{\left(x-2\right)^{2}}\times \frac{4-x+2x-6}{x-3}$$

Combine like terms in 4-x+2x-6.

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

Multiply \frac{x-3}{\left(x-2\right)^{2}} times \frac{-2+x}{x-3} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out \left(x-3\right)\left(x-2\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-5x+6}{x^{3}-6x^{2}+12x-8}.

$$\frac{\left(x-3\right)\left(x-2\right)}{\left(x-2\right)^{3}}\left(\frac{4-x}{x-3}+2\right)$$

Cancel out x-2 in both numerator and denominator.

$$\frac{x-3}{\left(x-2\right)^{2}}\left(\frac{4-x}{x-3}+2\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x-3}{x-3}.

$$\frac{x-3}{\left(x-2\right)^{2}}\left(\frac{4-x}{x-3}+\frac{2\left(x-3\right)}{x-3}\right)$$

Since \frac{4-x}{x-3} and \frac{2\left(x-3\right)}{x-3} have the same denominator, add them by adding their numerators.

$$\frac{x-3}{\left(x-2\right)^{2}}\times \frac{4-x+2\left(x-3\right)}{x-3}$$

Do the multiplications in 4-x+2\left(x-3\right).

$$\frac{x-3}{\left(x-2\right)^{2}}\times \frac{4-x+2x-6}{x-3}$$

Combine like terms in 4-x+2x-6.

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

Multiply \frac{x-3}{\left(x-2\right)^{2}} times \frac{-2+x}{x-3} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out \left(x-3\right)\left(x-2\right) in both numerator and denominator.

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