Solve a+1/a^2-a-1-a/a^2+a

$$\frac { a + 1 } { a ^ { 2 } - a } - \frac { 1 - a } { a ^ { 2 } + a }$$

Evaluate

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

Short Solution Steps

Factor a^{2}-a. Factor a^{2}+a.

$$\frac{a+1}{a\left(a-1\right)}-\frac{1-a}{a\left(a+1\right)}$$

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

$$\frac{\left(a+1\right)\left(a+1\right)}{a\left(a-1\right)\left(a+1\right)}-\frac{\left(1-a\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)}$$

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

$$\frac{\left(a+1\right)\left(a+1\right)-\left(1-a\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)}$$

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

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

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

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

Expand a\left(a-1\right)\left(a+1\right).

$$\frac{2a^{2}+2}{a^{3}-a}$$

Expand

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

Short Solution Steps

Factor a^{2}-a. Factor a^{2}+a.

$$\frac{a+1}{a\left(a-1\right)}-\frac{1-a}{a\left(a+1\right)}$$

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

$$\frac{\left(a+1\right)\left(a+1\right)}{a\left(a-1\right)\left(a+1\right)}-\frac{\left(1-a\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)}$$

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

$$\frac{\left(a+1\right)\left(a+1\right)-\left(1-a\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)}$$

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

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

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

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

Expand a\left(a-1\right)\left(a+1\right).

$$\frac{2a^{2}+2}{a^{3}-a}$$