Basic Math Examples

$\frac{r - 6}{r^{2} - 16} \div \frac{2 r - 12}{r^{2} + 3 r - 28}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{r - 6}{r^{2} - 16} \cdot \frac{r^{2} + 3 r - 28}{2 r - 12}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{r - 6}{r^{2} - 4^{2}} \cdot \frac{r^{2} + 3 r - 28}{2 r - 12}$

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 $a = r$ and $b = 4$ .

$\frac{r - 6}{(r + 4) (r - 4)} \cdot \frac{r^{2} + 3 r - 28}{2 r - 12}$

Step 3

Factor $r^{2} + 3 r - 28$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 3.2

Write the factored form using these integers.

$\frac{r - 6}{(r + 4) (r - 4)} \cdot \frac{(r - 4) (r + 7)}{2 r - 12}$

Step 4

Simplify terms.

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Step 4.1

Factor $2$ out of $2 r - 12$ .

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Step 4.1.1

Factor $2$ out of $2 r$ .

$\frac{r - 6}{(r + 4) (r - 4)} \cdot \frac{(r - 4) (r + 7)}{2 (r) - 12}$

Step 4.1.2

Factor $2$ out of $- 12$ .

$\frac{r - 6}{(r + 4) (r - 4)} \cdot \frac{(r - 4) (r + 7)}{2 r + 2 \cdot - 6}$

Step 4.1.3

Factor $2$ out of $2 r + 2 \cdot - 6$ .

$\frac{r - 6}{(r + 4) (r - 4)} \cdot \frac{(r - 4) (r + 7)}{2 (r - 6)}$

Step 4.2

Cancel the common factor of $r - 6$ .

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Step 4.2.1

Factor $r - 6$ out of $2 (r - 6)$ .

$\frac{r - 6}{(r + 4) (r - 4)} \cdot \frac{(r - 4) (r + 7)}{(r - 6) \cdot 2}$

Step 4.2.2

Cancel the common factor.

$\frac{r - 6}{(r + 4) (r - 4)} \cdot \frac{(r - 4) (r + 7)}{(r - 6) \cdot 2}$

Step 4.2.3

Rewrite the expression.

$\frac{1}{(r + 4) (r - 4)} \cdot \frac{(r - 4) (r + 7)}{2}$

Step 4.3

Cancel the common factor of $r - 4$ .

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Step 4.3.1

Factor $r - 4$ out of $(r + 4) (r - 4)$ .

$\frac{1}{(r - 4) (r + 4)} \cdot \frac{(r - 4) (r + 7)}{2}$

Step 4.3.2

Cancel the common factor.

$\frac{1}{(r - 4) (r + 4)} \cdot \frac{(r - 4) (r + 7)}{2}$

Step 4.3.3

Rewrite the expression.

$\frac{1}{r + 4} \cdot \frac{r + 7}{2}$

Step 4.4

Multiply $\frac{1}{r + 4}$ by $\frac{r + 7}{2}$ .

$\frac{r + 7}{(r + 4) \cdot 2}$

Step 4.5

Move $2$ to the left of $r + 4$ .

$\frac{r + 7}{2 (r + 4)}$