Algebra Examples

$\frac{x^{2} + 8 x + 16}{x + 3} \div \frac{2 x + 8}{x^{2} - 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 8 x + 16}{x + 3} \cdot \frac{x^{2} - 9}{2 x + 8}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{x^{2} + 8 x + 4^{2}}{x + 3} \cdot \frac{x^{2} - 9}{2 x + 8}$

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.

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

Step 2.3

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 4 + 4^{2}}{x + 3} \cdot \frac{x^{2} - 9}{2 x + 8}$

Step 2.4

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

$\frac{{(x + 4)}^{2}}{x + 3} \cdot \frac{x^{2} - 9}{2 x + 8}$

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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 $b = 3$ .

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

Step 4

Simplify terms.

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Step 4.1

Factor $2$ out of $2 x + 8$ .

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Step 4.1.1

Factor $2$ out of $2 x$ .

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

Step 4.1.2

Factor $2$ out of $8$ .

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

Step 4.1.3

Factor $2$ out of $2 x + 2 \cdot 4$ .

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

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of $x + 4$ .

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Cancel the common factor.

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

Step 4.2.1.4

Rewrite the expression.

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

Step 4.2.2

Cancel the common factor of $x + 3$ .

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Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Rewrite the expression.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.2.4

Combine $x$ and $\frac{x - 3}{2}$ .

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

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Factor $2$ out of $4$ .

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

Step 4.2.5.2

Cancel the common factor.

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

Step 4.2.5.3

Rewrite the expression.

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

Step 4.3

Simplify each term.

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Step 4.3.1

Apply the distributive property.

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

Step 4.3.2

Multiply $2$ by $- 3$ .

$\frac{x (x - 3)}{2} + 2 x - 6$

Step 5

Find the common denominator.

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Step 5.1

Write $2 x$ as a fraction with denominator $1$ .

$\frac{x (x - 3)}{2} + \frac{2 x}{1} - 6$

Step 5.2

Multiply $\frac{2 x}{1}$ by $\frac{2}{2}$ .

$\frac{x (x - 3)}{2} + \frac{2 x}{1} \cdot \frac{2}{2} - 6$

Step 5.3

Multiply $\frac{2 x}{1}$ by $\frac{2}{2}$ .

$\frac{x (x - 3)}{2} + \frac{2 x \cdot 2}{2} - 6$

Step 5.4

Write $- 6$ as a fraction with denominator $1$ .

$\frac{x (x - 3)}{2} + \frac{2 x \cdot 2}{2} + \frac{- 6}{1}$

Step 5.5

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$\frac{x (x - 3)}{2} + \frac{2 x \cdot 2}{2} + \frac{- 6}{1} \cdot \frac{2}{2}$

Step 5.6

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$\frac{x (x - 3)}{2} + \frac{2 x \cdot 2}{2} + \frac{- 6 \cdot 2}{2}$

Step 6

Simplify terms.

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Step 6.1

Combine the numerators over the common denominator.

$\frac{x (x - 3) + 2 x \cdot 2 - 6 \cdot 2}{2}$

Step 6.2

Simplify each term.

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Step 6.2.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 + 2 x \cdot 2 - 6 \cdot 2}{2}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 3 + 2 x \cdot 2 - 6 \cdot 2}{2}$

Step 6.2.3

Move $- 3$ to the left of $x$ .

$\frac{x^{2} - 3 \cdot x + 2 x \cdot 2 - 6 \cdot 2}{2}$

Step 6.2.4

Multiply $2$ by $2$ .

$\frac{x^{2} - 3 x + 4 x - 6 \cdot 2}{2}$

Step 6.2.5

Multiply $- 6$ by $2$ .

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

Step 6.3

Add $- 3 x$ and $4 x$ .

$\frac{x^{2} + x - 12}{2}$

Step 7

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

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Step 7.1

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 $1$ .

$- 3, 4$

Step 7.2

Write the factored form using these integers.

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