Algebra Examples

$\frac{v^{2} - 9}{3 v^{2} + 12} \div \frac{v + 3}{v^{2} + 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{v^{2} - 9}{3 v^{2} + 12} \cdot \frac{v^{2} + 4}{v + 3}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{v^{2} - 3^{2}}{3 v^{2} + 12} \cdot \frac{v^{2} + 4}{v + 3}$

Step 2.2

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

$\frac{(v + 3) (v - 3)}{3 v^{2} + 12} \cdot \frac{v^{2} + 4}{v + 3}$

Step 3

Factor $3$ out of $3 v^{2} + 12$ .

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Step 3.1

Factor $3$ out of $3 v^{2}$ .

$\frac{(v + 3) (v - 3)}{3 (v^{2}) + 12} \cdot \frac{v^{2} + 4}{v + 3}$

Step 3.2

Factor $3$ out of $12$ .

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

Step 3.3

Factor $3$ out of $3 v^{2} + 3 \cdot 4$ .

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

Step 4

Cancel the common factor of $v + 3$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\frac{v - 3}{3 (v^{2} + 4)} (v^{2} + 4)$

Step 5

Cancel the common factor of $v^{2} + 4$ .

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Step 5.1

Factor $v^{2} + 4$ out of $3 (v^{2} + 4)$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

$\frac{v - 3}{3}$

Step 6

Split the fraction $\frac{v - 3}{3}$ into two fractions.

$\frac{v}{3} + \frac{- 3}{3}$

Step 7

Divide $- 3$ by $3$ .

$\frac{v}{3} - 1$