Simplify t/(t+3)+(4t)/(t-3)-18/(t^2-9)

$\frac{t}{t + 3} + \frac{4 t}{t - 3} - \frac{18}{t^{2} - 9}$

Simplify the denominator.

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

$\frac{t}{t + 3} + \frac{4 t}{t - 3} - \frac{18}{t^{2} - 3^{2}}$

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

$\frac{t}{t + 3} + \frac{4 t}{t - 3} - \frac{18}{(t + 3) (t - 3)}$

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

$\frac{t}{t + 3} \cdot \frac{t - 3}{t - 3} + \frac{4 t}{t - 3} - \frac{18}{(t + 3) (t - 3)}$

To write $\frac{4 t}{t - 3}$ as a fraction with a common denominator, multiply by $\frac{t + 3}{t + 3}$ .

$\frac{t}{t + 3} \cdot \frac{t - 3}{t - 3} + \frac{4 t}{t - 3} \cdot \frac{t + 3}{t + 3} - \frac{18}{(t + 3) (t - 3)}$

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

Multiply $\frac{t}{t + 3}$ by $\frac{t - 3}{t - 3}$ .

$\frac{t (t - 3)}{(t + 3) (t - 3)} + \frac{4 t}{t - 3} \cdot \frac{t + 3}{t + 3} - \frac{18}{(t + 3) (t - 3)}$

Multiply $\frac{4 t}{t - 3}$ by $\frac{t + 3}{t + 3}$ .

$\frac{t (t - 3)}{(t + 3) (t - 3)} + \frac{4 t (t + 3)}{(t - 3) (t + 3)} - \frac{18}{(t + 3) (t - 3)}$

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

$\frac{t (t - 3)}{(t + 3) (t - 3)} + \frac{4 t (t + 3)}{(t + 3) (t - 3)} - \frac{18}{(t + 3) (t - 3)}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{t (t - 3) + 4 t (t + 3)}{(t + 3) (t - 3)} - \frac{18}{(t + 3) (t - 3)}$

Combine the numerators over the common denominator.

$\frac{t (t - 3) + 4 t (t + 3) - 18}{(t + 3) (t - 3)}$

Simplify each term.

Apply the distributive property.

$\frac{t \cdot t + t \cdot - 3 + 4 t (t + 3) - 18}{(t + 3) (t - 3)}$

Multiply $t$ by $t$ .

$\frac{t^{2} + t \cdot - 3 + 4 t (t + 3) - 18}{(t + 3) (t - 3)}$

Move $- 3$ to the left of $t$ .

$\frac{t^{2} - 3 \cdot t + 4 t (t + 3) - 18}{(t + 3) (t - 3)}$

Apply the distributive property.

$\frac{t^{2} - 3 t + 4 t \cdot t + 4 t \cdot 3 - 18}{(t + 3) (t - 3)}$

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$\frac{t^{2} - 3 t + 4 (t \cdot t) + 4 t \cdot 3 - 18}{(t + 3) (t - 3)}$

Multiply $t$ by $t$ .

$\frac{t^{2} - 3 t + 4 t^{2} + 4 t \cdot 3 - 18}{(t + 3) (t - 3)}$

Multiply $3$ by $4$ .

$\frac{t^{2} - 3 t + 4 t^{2} + 12 t - 18}{(t + 3) (t - 3)}$

Simplify by adding terms.

Add $t^{2}$ and $4 t^{2}$ .

$\frac{5 t^{2} - 3 t + 12 t - 18}{(t + 3) (t - 3)}$

Add $- 3 t$ and $12 t$ .

$\frac{5 t^{2} + 9 t - 18}{(t + 3) (t - 3)}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 18 = - 90$ and whose sum is $b = 9$ .

Factor $9$ out of $9 t$ .

$\frac{5 t^{2} + 9 (t) - 18}{(t + 3) (t - 3)}$

Rewrite $9$ as $- 6$ plus $15$

$\frac{5 t^{2} + (- 6 + 15) t - 18}{(t + 3) (t - 3)}$

Apply the distributive property.

$\frac{5 t^{2} - 6 t + 15 t - 18}{(t + 3) (t - 3)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(5 t^{2} - 6 t) + 15 t - 18}{(t + 3) (t - 3)}$

Factor out the greatest common factor (GCF) from each group.

$\frac{t (5 t - 6) + 3 (5 t - 6)}{(t + 3) (t - 3)}$

Factor the polynomial by factoring out the greatest common factor, $5 t - 6$ .

$\frac{(5 t - 6) (t + 3)}{(t + 3) (t - 3)}$

Cancel the common factor of $t + 3$ .

Cancel the common factor.

$\frac{(5 t - 6) (t + 3)}{(t + 3) (t - 3)}$

Rewrite the expression.

$\frac{5 t - 6}{t - 3}$

Simplify t/(t+3)+(4t)/(t-3)-18/(t^2-9)

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