Simplify (2a)/(a-3)-(2a)/(a+3)+36/(a^2-9)

$\frac{2 a}{a - 3} - \frac{2 a}{a + 3} + \frac{36}{a^{2} - 9}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{2 a}{a - 3} - \frac{2 a}{a + 3} + \frac{36}{a^{2} - 3^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 3$ .

$\frac{2 a}{a - 3} - \frac{2 a}{a + 3} + \frac{36}{(a + 3) (a - 3)}$

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

$\frac{2 a}{a - 3} \cdot \frac{a + 3}{a + 3} - \frac{2 a}{a + 3} + \frac{36}{(a + 3) (a - 3)}$

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

$\frac{2 a}{a - 3} \cdot \frac{a + 3}{a + 3} - \frac{2 a}{a + 3} \cdot \frac{a - 3}{a - 3} + \frac{36}{(a + 3) (a - 3)}$

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

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

$\frac{2 a (a + 3)}{(a - 3) (a + 3)} - \frac{2 a}{a + 3} \cdot \frac{a - 3}{a - 3} + \frac{36}{(a + 3) (a - 3)}$

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

$\frac{2 a (a + 3)}{(a - 3) (a + 3)} - \frac{2 a (a - 3)}{(a + 3) (a - 3)} + \frac{36}{(a + 3) (a - 3)}$

Reorder the factors of $(a - 3) (a + 3)$ .

$\frac{2 a (a + 3)}{(a + 3) (a - 3)} - \frac{2 a (a - 3)}{(a + 3) (a - 3)} + \frac{36}{(a + 3) (a - 3)}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{2 a (a + 3) - 2 a (a - 3)}{(a + 3) (a - 3)} + \frac{36}{(a + 3) (a - 3)}$

Combine the numerators over the common denominator.

$\frac{2 a (a + 3) - 2 a (a - 3) + 36}{(a + 3) (a - 3)}$

Simplify each term.

Apply the distributive property.

$\frac{2 a \cdot a + 2 a \cdot 3 - 2 a (a - 3) + 36}{(a + 3) (a - 3)}$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$\frac{2 (a \cdot a) + 2 a \cdot 3 - 2 a (a - 3) + 36}{(a + 3) (a - 3)}$

Multiply $a$ by $a$ .

$\frac{2 a^{2} + 2 a \cdot 3 - 2 a (a - 3) + 36}{(a + 3) (a - 3)}$

Multiply $3$ by $2$ .

$\frac{2 a^{2} + 6 a - 2 a (a - 3) + 36}{(a + 3) (a - 3)}$

Apply the distributive property.

$\frac{2 a^{2} + 6 a - 2 a \cdot a - 2 a \cdot - 3 + 36}{(a + 3) (a - 3)}$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$\frac{2 a^{2} + 6 a - 2 (a \cdot a) - 2 a \cdot - 3 + 36}{(a + 3) (a - 3)}$

Multiply $a$ by $a$ .

$\frac{2 a^{2} + 6 a - 2 a^{2} - 2 a \cdot - 3 + 36}{(a + 3) (a - 3)}$

Multiply $- 3$ by $- 2$ .

$\frac{2 a^{2} + 6 a - 2 a^{2} + 6 a + 36}{(a + 3) (a - 3)}$

Simplify terms.

Combine the opposite terms in $2 a^{2} + 6 a - 2 a^{2} + 6 a + 36$ .

Subtract $2 a^{2}$ from $2 a^{2}$ .

$\frac{6 a + 0 + 6 a + 36}{(a + 3) (a - 3)}$

Add $6 a$ and $0$ .

$\frac{6 a + 6 a + 36}{(a + 3) (a - 3)}$

Add $6 a$ and $6 a$ .

$\frac{12 a + 36}{(a + 3) (a - 3)}$

Factor $12$ out of $12 a + 36$ .

Factor $12$ out of $12 a$ .

$\frac{12 (a) + 36}{(a + 3) (a - 3)}$

Factor $12$ out of $36$ .

$\frac{12 a + 12 \cdot 3}{(a + 3) (a - 3)}$

Factor $12$ out of $12 a + 12 \cdot 3$ .

$\frac{12 (a + 3)}{(a + 3) (a - 3)}$

Cancel the common factor of $a + 3$ .

Cancel the common factor.

$\frac{12 (a + 3)}{(a + 3) (a - 3)}$

Rewrite the expression.

$\frac{12}{a - 3}$

Simplify (2a)/(a-3)-(2a)/(a+3)+36/(a^2-9)

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