Solve (2)x-2/x-1*x^2-1/x^2-4x+4-1/x-2

$$( 2 ) \frac { x - 2 } { x - 1 } \cdot \frac { x ^ { 2 } - 1 } { x ^ { 2 } - 4 x + 4 } - \frac { 1 } { x - 2 }$$

Evaluate

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

Short Solution Steps

Express 2\times \frac{x-2}{x-1} as a single fraction.

$$\frac{2\left(x-2\right)}{x-1}\times \frac{x^{2}-1}{x^{2}-4x+4}-\frac{1}{x-2}$$

Multiply \frac{2\left(x-2\right)}{x-1} times \frac{x^{2}-1}{x^{2}-4x+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\left(x-2\right)\left(x^{2}-1\right)}{\left(x-1\right)\left(x^{2}-4x+4\right)}-\frac{1}{x-2}$$

Factor \left(x-1\right)\left(x^{2}-4x+4\right).

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

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

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

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

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

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

$$\frac{2x^{3}-2x-4x^{2}+4-x^{2}+x+2x-2}{\left(x-1\right)\left(x-2\right)^{2}}$$

Combine like terms in 2x^{3}-2x-4x^{2}+4-x^{2}+x+2x-2.

$$\frac{2x^{3}+x-5x^{2}+2}{\left(x-1\right)\left(x-2\right)^{2}}$$

Factor the expressions that are not already factored in \frac{2x^{3}+x-5x^{2}+2}{\left(x-1\right)\left(x-2\right)^{2}}.

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

Cancel out x-1 in both numerator and denominator.

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

Expand \left(x-2\right)^{2}.

$$\frac{2\left(\frac{1}{2}x-1\right)\left(2x+1\right)}{x^{2}-4x+4}$$

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Express 2\times \frac{x-2}{x-1} as a single fraction.

$$\frac{2\left(x-2\right)}{x-1}\times \frac{x^{2}-1}{x^{2}-4x+4}-\frac{1}{x-2}$$

Multiply \frac{2\left(x-2\right)}{x-1} times \frac{x^{2}-1}{x^{2}-4x+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\left(x-2\right)\left(x^{2}-1\right)}{\left(x-1\right)\left(x^{2}-4x+4\right)}-\frac{1}{x-2}$$

Factor \left(x-1\right)\left(x^{2}-4x+4\right).

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

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

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

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

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

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

$$\frac{2x^{3}-2x-4x^{2}+4-x^{2}+x+2x-2}{\left(x-1\right)\left(x-2\right)^{2}}$$

Combine like terms in 2x^{3}-2x-4x^{2}+4-x^{2}+x+2x-2.

$$\frac{2x^{3}+x-5x^{2}+2}{\left(x-1\right)\left(x-2\right)^{2}}$$

Factor the expressions that are not already factored in \frac{2x^{3}+x-5x^{2}+2}{\left(x-1\right)\left(x-2\right)^{2}}.

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

Cancel out x-1 in both numerator and denominator.

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

Expand \left(x-2\right)^{2}.

$$\frac{2\left(\frac{1}{2}x-1\right)\left(2x+1\right)}{x^{2}-4x+4}$$

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

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

Expand the expression.

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