Algebra Examples

$\frac{x + 2}{x - 3} + \frac{x + 2}{x + 5}$

Step 1

To write $\frac{x + 2}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{x + 2}{x - 3} \cdot \frac{x + 5}{x + 5} + \frac{x + 2}{x + 5}$

Step 2

To write $\frac{x + 2}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x + 2}{x - 3} \cdot \frac{x + 5}{x + 5} + \frac{x + 2}{x + 5} \cdot \frac{x - 3}{x - 3}$

Step 3

Write each expression with a common denominator of $(x - 3) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x + 2}{x - 3}$ by $\frac{x + 5}{x + 5}$ .

$\frac{(x + 2) (x + 5)}{(x - 3) (x + 5)} + \frac{x + 2}{x + 5} \cdot \frac{x - 3}{x - 3}$

Step 3.2

Multiply $\frac{x + 2}{x + 5}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x + 2) (x + 5)}{(x - 3) (x + 5)} + \frac{(x + 2) (x - 3)}{(x + 5) (x - 3)}$

Step 3.3

Reorder the factors of $(x - 3) (x + 5)$ .

$\frac{(x + 2) (x + 5)}{(x + 5) (x - 3)} + \frac{(x + 2) (x - 3)}{(x + 5) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 2) (x + 5) + (x + 2) (x - 3)}{(x + 5) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $x + 2$ out of $(x + 2) (x + 5) + (x + 2) (x - 3)$ .

$\frac{(x + 2) (x + 5 + x - 3)}{(x + 5) (x - 3)}$

Step 5.2

Add $x$ and $x$ .

$\frac{(x + 2) (2 x + 5 - 3)}{(x + 5) (x - 3)}$

Step 5.3

Subtract $3$ from $5$ .

$\frac{(x + 2) (2 x + 2)}{(x + 5) (x - 3)}$

Step 5.4

Factor $2$ out of $2 x + 2$ .

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Step 5.4.1

Factor $2$ out of $2 x$ .

$\frac{(x + 2) (2 (x) + 2)}{(x + 5) (x - 3)}$

Step 5.4.2

Factor $2$ out of $2$ .

$\frac{(x + 2) (2 (x) + 2 (1))}{(x + 5) (x - 3)}$

Step 5.4.3

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

$\frac{(x + 2) (2 (x + 1))}{(x + 5) (x - 3)}$

$\frac{(x + 2) \cdot 2 (x + 1)}{(x + 5) (x - 3)}$

Step 6

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

$\frac{2 (x + 2) (x + 1)}{(x + 5) (x - 3)}$