Algebra Examples

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

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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 $b = 4$ .

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

Step 2

Factor using the perfect square rule.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 3

Factor $x^{2} + 6 x + 8$ 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 $8$ and whose sum is $6$ .

$2, 4$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Cancel the common factor of $x + 4$ .

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 2$ .

Tap for more steps...

Step 4.2.1

Factor $x - 2$ out of ${(x - 2)}^{2}$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

Multiply $\frac{x - 4}{x - 2}$ by $\frac{1}{x + 2}$ .

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