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

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

Evaluate

$$\frac{3}{a^{2}-1}$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out a in both numerator and denominator.

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

Factor a^{2}-1.

$$\frac{4}{\left(a-1\right)\left(a+1\right)}-\frac{1}{\left(a-1\right)\left(a+1\right)}$$

Since \frac{4}{\left(a-1\right)\left(a+1\right)} and \frac{1}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators. Subtract 1 from 4 to get 3.

$$\frac{3}{\left(a-1\right)\left(a+1\right)}$$

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

$$\frac{3}{a^{2}-1}$$

Expand

$$\frac{3}{a^{2}-1}$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out a in both numerator and denominator.

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

Factor a^{2}-1.

$$\frac{4}{\left(a-1\right)\left(a+1\right)}-\frac{1}{\left(a-1\right)\left(a+1\right)}$$

Since \frac{4}{\left(a-1\right)\left(a+1\right)} and \frac{1}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators. Subtract 1 from 4 to get 3.

$$\frac{3}{\left(a-1\right)\left(a+1\right)}$$

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

$$\frac{3}{a^{2}-1}$$