Trigonometry Examples

$\frac{2 x}{2 x - 2} + \frac{2}{x + 2}$

Step 1

Cancel the common factor of $2$ and $2 x - 2$ .

Tap for more steps...

Step 1.1

Factor $2$ out of $2 x$ .

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

Step 1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.1

Factor $2$ out of $2 x$ .

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

Step 1.2.2

Factor $2$ out of $- 2$ .

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

Step 1.2.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

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

Step 1.2.4

Cancel the common factor.

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

Step 1.2.5

Rewrite the expression.

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

Step 2

To write $\frac{x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 3

To write $\frac{2}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 4

Write each expression with a common denominator of $(x - 1) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{x}{x - 1}$ by $\frac{x + 2}{x + 2}$ .

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

Step 4.2

Multiply $\frac{2}{x + 2}$ by $\frac{x - 1}{x - 1}$ .

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

Step 4.3

Reorder the factors of $(x - 1) (x + 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 6.3

Move $2$ to the left of $x$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $2$ by $- 1$ .

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

Step 6.6

Add $2 x$ and $2 x$ .

$\frac{x^{2} + 4 x - 2}{(x + 2) (x - 1)}$