Simplify (2x)/(x^2-36)+(x+4)/(x+6)

$\frac{2 x}{x^{2} - 36} + \frac{x + 4}{x + 6}$

Step 1

Simplify the denominator.

Rewrite $36$ as $6^{2}$ .

$\frac{2 x}{x^{2} - 6^{2}} + \frac{x + 4}{x + 6}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$\frac{2 x}{(x + 6) (x - 6)} + \frac{x + 4}{x + 6}$

Step 2

To write $\frac{x + 4}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{2 x}{(x + 6) (x - 6)} + \frac{x + 4}{x + 6} \cdot \frac{x - 6}{x - 6}$

Step 3

Simplify terms.

Multiply $\frac{x + 4}{x + 6}$ by $\frac{x - 6}{x - 6}$ .

$\frac{2 x}{(x + 6) (x - 6)} + \frac{(x + 4) (x - 6)}{(x + 6) (x - 6)}$

Combine the numerators over the common denominator.

$\frac{2 x + (x + 4) (x - 6)}{(x + 6) (x - 6)}$

Step 4

Simplify the numerator.

Expand $(x + 4) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 x + x (x - 6) + 4 (x - 6)}{(x + 6) (x - 6)}$

Apply the distributive property.

$\frac{2 x + x \cdot x + x \cdot - 6 + 4 (x - 6)}{(x + 6) (x - 6)}$

Apply the distributive property.

$\frac{2 x + x \cdot x + x \cdot - 6 + 4 x + 4 \cdot - 6}{(x + 6) (x - 6)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{2 x + x^{2} + x \cdot - 6 + 4 x + 4 \cdot - 6}{(x + 6) (x - 6)}$

Move $- 6$ to the left of $x$ .

$\frac{2 x + x^{2} - 6 \cdot x + 4 x + 4 \cdot - 6}{(x + 6) (x - 6)}$

Multiply $4$ by $- 6$ .

$\frac{2 x + x^{2} - 6 x + 4 x - 24}{(x + 6) (x - 6)}$

Add $- 6 x$ and $4 x$ .

$\frac{2 x + x^{2} - 2 x - 24}{(x + 6) (x - 6)}$

Subtract $2 x$ from $2 x$ .

$\frac{x^{2} + 0 - 24}{(x + 6) (x - 6)}$

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

$\frac{x^{2} - 24}{(x + 6) (x - 6)}$