Solve a+1/a-1+a+1/a

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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