Algebra Examples

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

Step 1

Find the common denominator.

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Step 1.1

Write $a$ as a fraction with denominator $1$ .

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

Step 1.2

Multiply $\frac{a}{1}$ by $\frac{a - 1}{a - 1}$ .

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

Step 1.3

Multiply $\frac{a}{1}$ by $\frac{a - 1}{a - 1}$ .

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

Step 1.4

Write $2$ as a fraction with denominator $1$ .

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

Step 1.5

Multiply $\frac{2}{1}$ by $\frac{a - 1}{a - 1}$ .

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

Step 1.6

Multiply $\frac{2}{1}$ by $\frac{a - 1}{a - 1}$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $a$ by $a$ .

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

Step 3.3

Move $- 1$ to the left of $a$ .

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

Step 3.4

Rewrite $- 1 a$ as $- a$ .

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

Step 3.5

Apply the distributive property.

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

Step 3.6

Multiply $2$ by $- 1$ .

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

Step 4

Combine the opposite terms in $a + a^{2} - a + 2 a - 2$ .

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Step 4.1

Subtract $a$ from $a$ .

$\frac{a^{2} + 0 + 2 a - 2}{a - 1}$

Step 4.2

Add $a^{2}$ and $0$ .

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