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

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

Evaluate

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

Short Solution Steps

Factor x^{2}-1.

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

Factor x^{2}-1.

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

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

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

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

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

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

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

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

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

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

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