Algebra Examples

$\frac{x^{2} - x - 12}{x + 4} \div \frac{x^{2} - 8 x + 16}{2 x^{2} - 32}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - x - 12}{x + 4} \cdot \frac{2 x^{2} - 32}{x^{2} - 8 x + 16}$

Step 2

Factor $x^{2} - x - 12$ using the AC method.

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Step 2.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 3)}{x + 4} \cdot \frac{2 x^{2} - 32}{x^{2} - 8 x + 16}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $2$ out of $2 x^{2} - 32$ .

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Step 3.1.1

Factor $2$ out of $2 x^{2}$ .

$\frac{(x - 4) (x + 3)}{x + 4} \cdot \frac{2 (x^{2}) - 32}{x^{2} - 8 x + 16}$

Step 3.1.2

Factor $2$ out of $- 32$ .

$\frac{(x - 4) (x + 3)}{x + 4} \cdot \frac{2 x^{2} + 2 \cdot - 16}{x^{2} - 8 x + 16}$

Step 3.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 16$ .

$\frac{(x - 4) (x + 3)}{x + 4} \cdot \frac{2 (x^{2} - 16)}{x^{2} - 8 x + 16}$

Step 3.2

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

$\frac{(x - 4) (x + 3)}{x + 4} \cdot \frac{2 (x^{2} - 4^{2})}{x^{2} - 8 x + 16}$

Step 3.3

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 $b = 4$ .

$\frac{(x - 4) (x + 3)}{x + 4} \cdot \frac{2 (x + 4) (x - 4)}{x^{2} - 8 x + 16}$

Step 4

Factor using the perfect square rule.

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Step 4.1

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

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

Step 4.2

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

$8 x = 2 \cdot x \cdot 4$

Step 4.3

Rewrite the polynomial.

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

Step 4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 5

Cancel the common factor of $x - 4$ .

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Step 5.1

Factor $x - 4$ out of ${(x - 4)}^{2}$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $x + 4$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Cancel the common factor of $x - 4$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Divide $2$ by $1$ .

$(x + 3) \cdot 2$

Step 8

Apply the distributive property.

$x \cdot 2 + 3 \cdot 2$

Step 9

Move $2$ to the left of $x$ .

$2 \cdot x + 3 \cdot 2$

Step 10

Multiply $3$ by $2$ .

$2 x + 6$