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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Expand \left(x-1\right)^{2}.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

Expand \left(x-1\right)^{2}.

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