Algebra Examples

$\frac{\frac{x^{2} - 16}{x^{2} + 5 x + 6}}{\frac{x^{2} + 5 x + 4}{x^{2} - 2 x - 8}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 16}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 2 x - 8}{x^{2} + 5 x + 4}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

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

$\frac{x^{2} - 4^{2}}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 2 x - 8}{x^{2} + 5 x + 4}$

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 $a = x$ and $b = 4$ .

$\frac{(x + 4) (x - 4)}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 2 x - 8}{x^{2} + 5 x + 4}$

Step 3

Factor $x^{2} + 5 x + 6$ using the AC method.

Tap for more steps...

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 $6$ and whose sum is $5$ .

$2, 3$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 4) (x - 4)}{(x + 2) (x + 3)} \cdot \frac{x^{2} - 2 x - 8}{x^{2} + 5 x + 4}$

Step 4

Factor $x^{2} - 2 x - 8$ using the AC method.

Tap for more steps...

Step 4.1

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 4) (x - 4)}{(x + 2) (x + 3)} \cdot \frac{(x - 4) (x + 2)}{x^{2} + 5 x + 4}$

Step 5

Factor $x^{2} + 5 x + 4$ using the AC method.

Tap for more steps...

Step 5.1

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 $4$ and whose sum is $5$ .

$1, 4$

Step 5.2

Write the factored form using these integers.

$\frac{(x + 4) (x - 4)}{(x + 2) (x + 3)} \cdot \frac{(x - 4) (x + 2)}{(x + 1) (x + 4)}$

Step 6

Cancel the common factor of $x + 4$ .

Tap for more steps...

Step 6.1

Factor $x + 4$ out of $(x + 1) (x + 4)$ .

$\frac{(x + 4) (x - 4)}{(x + 2) (x + 3)} \cdot \frac{(x - 4) (x + 2)}{(x + 4) (x + 1)}$

Step 6.2

Cancel the common factor.

$\frac{(x + 4) (x - 4)}{(x + 2) (x + 3)} \cdot \frac{(x - 4) (x + 2)}{(x + 4) (x + 1)}$

Step 6.3

Rewrite the expression.

$\frac{x - 4}{(x + 2) (x + 3)} \cdot \frac{(x - 4) (x + 2)}{x + 1}$

Step 7

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 7.1

Factor $x + 2$ out of $(x - 4) (x + 2)$ .

$\frac{x - 4}{(x + 2) (x + 3)} \cdot \frac{(x + 2) (x - 4)}{x + 1}$

Step 7.2

Cancel the common factor.

$\frac{x - 4}{(x + 2) (x + 3)} \cdot \frac{(x + 2) (x - 4)}{x + 1}$

Step 7.3

Rewrite the expression.

$\frac{x - 4}{x + 3} \cdot \frac{x - 4}{x + 1}$

Step 8

Multiply $\frac{x - 4}{x + 3}$ by $\frac{x - 4}{x + 1}$ .

$\frac{(x - 4) (x - 4)}{(x + 3) (x + 1)}$

Step 9

Raise $x - 4$ to the power of $1$ .

$\frac{{(x - 4)}^{1} (x - 4)}{(x + 3) (x + 1)}$

Step 10

Raise $x - 4$ to the power of $1$ .

$\frac{{(x - 4)}^{1} {(x - 4)}^{1}}{(x + 3) (x + 1)}$

Step 11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x - 4)}^{1 + 1}}{(x + 3) (x + 1)}$

Step 12

Add $1$ and $1$ .

$\frac{{(x - 4)}^{2}}{(x + 3) (x + 1)}$