Algebra Examples

$\frac{3 x^{2} - 192}{64 - x^{2}} \cdot \frac{4}{x^{2} + 16 x + 64} \div \frac{x + 10}{x^{2} + 18 x + 80}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} - 192}{64 - x^{2}} \cdot \frac{4}{x^{2} + 16 x + 64} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $3 x^{2} - 192$ .

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Step 2.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 192}{64 - x^{2}} \cdot \frac{4}{x^{2} + 16 x + 64} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 2.1.2

Factor $3$ out of $- 192$ .

$\frac{3 x^{2} + 3 \cdot - 64}{64 - x^{2}} \cdot \frac{4}{x^{2} + 16 x + 64} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 2.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 64$ .

$\frac{3 (x^{2} - 64)}{64 - x^{2}} \cdot \frac{4}{x^{2} + 16 x + 64} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 2.2

Rewrite $64$ as $8^{2}$ .

$\frac{3 (x^{2} - 8^{2})}{64 - x^{2}} \cdot \frac{4}{x^{2} + 16 x + 64} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 2.3

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 $b = 8$ .

$\frac{3 (x + 8) (x - 8)}{64 - x^{2}} \cdot \frac{4}{x^{2} + 16 x + 64} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $64$ as $8^{2}$ .

$\frac{3 (x + 8) (x - 8)}{8^{2} - x^{2}} \cdot \frac{4}{x^{2} + 16 x + 64} \frac{x^{2} + 18 x + 80}{x + 10}$

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 $a = 8$ and $b = x$ .

$\frac{3 (x + 8) (x - 8)}{(8 + x) (8 - x)} \cdot \frac{4}{x^{2} + 16 x + 64} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 4

Factor using the perfect square rule.

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Step 4.1

Rewrite $64$ as $8^{2}$ .

$\frac{3 (x + 8) (x - 8)}{(8 + x) (8 - x)} \cdot \frac{4}{x^{2} + 16 x + 8^{2}} \frac{x^{2} + 18 x + 80}{x + 10}$

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.

$16 x = 2 \cdot x \cdot 8$

Step 4.3

Rewrite the polynomial.

$\frac{3 (x + 8) (x - 8)}{(8 + x) (8 - x)} \cdot \frac{4}{x^{2} + 2 \cdot x \cdot 8 + 8^{2}} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 8$ .

$\frac{3 (x + 8) (x - 8)}{(8 + x) (8 - x)} \cdot \frac{4}{{(x + 8)}^{2}} \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{3 (x + 8) (x - 8) \cdot 4}{(8 + x) (8 - x) {(x + 8)}^{2}} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.2

Cancel the common factor of $x + 8$ and ${(x + 8)}^{2}$ .

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Step 5.2.1

Factor $x + 8$ out of $3 (x + 8) (x - 8) \cdot 4$ .

$\frac{(x + 8) ((3 (x - 8)) \cdot 4)}{(8 + x) (8 - x) {(x + 8)}^{2}} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $x + 8$ out of $(8 + x) (8 - x) {(x + 8)}^{2}$ .

$\frac{(x + 8) ((3 (x - 8)) \cdot 4)}{(x + 8) ((8 + x) (8 - x) (x + 8))} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.2.2.2

Cancel the common factor.

$\frac{(x + 8) ((3 (x - 8)) \cdot 4)}{(x + 8) ((8 + x) (8 - x) (x + 8))} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.2.2.3

Rewrite the expression.

$\frac{(3 (x - 8)) \cdot 4}{(8 + x) (8 - x) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.3

Cancel the common factor of $x - 8$ and $8 - x$ .

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Step 5.3.1

Factor $- 1$ out of $x$ .

$\frac{3 (- 1 (- x) - 8) \cdot 4}{(8 + x) (8 - x) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.3.2

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{3 (- 1 (- x) - 1 (8)) \cdot 4}{(8 + x) (8 - x) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.3.3

Factor $- 1$ out of $- 1 (- x) - 1 (8)$ .

$\frac{3 (- 1 (- x + 8)) \cdot 4}{(8 + x) (8 - x) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.3.4

Reorder terms.

$\frac{3 (- 1 (- x + 8)) \cdot 4}{(8 + x) (- x + 8) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.3.5

Cancel the common factor.

$\frac{3 (- 1 (- x + 8)) \cdot 4}{(8 + x) (- x + 8) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 5.3.6

Rewrite the expression.

$\frac{3 \cdot (- 1) \cdot 4}{(8 + x) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $3$ by $- 1$ .

$\frac{- 3 \cdot 4}{(8 + x) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 6.2

Multiply $- 3$ by $4$ .

$\frac{- 12}{(8 + x) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 7

Simplify the denominator.

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Step 7.1

Reorder terms.

$\frac{- 12}{(x + 8) (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 7.2

Raise $x + 8$ to the power of $1$ .

$\frac{- 12}{{(x + 8)}^{1} (x + 8)} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 7.3

Raise $x + 8$ to the power of $1$ .

$\frac{- 12}{{(x + 8)}^{1} {(x + 8)}^{1}} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 7.4

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

$\frac{- 12}{{(x + 8)}^{1 + 1}} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 7.5

Add $1$ and $1$ .

$\frac{- 12}{{(x + 8)}^{2}} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 8

Move the negative in front of the fraction.

$- \frac{12}{{(x + 8)}^{2}} \cdot \frac{x^{2} + 18 x + 80}{x + 10}$

Step 9

Factor $x^{2} + 18 x + 80$ using the AC method.

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Step 9.1

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 $80$ and whose sum is $18$ .

$8, 10$

Step 9.2

Write the factored form using these integers.

$- \frac{12}{{(x + 8)}^{2}} \cdot \frac{(x + 8) (x + 10)}{x + 10}$

Step 10

Simplify terms.

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Step 10.1

Cancel the common factor of $x + 8$ .

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Step 10.1.1

Move the leading negative in $- \frac{12}{{(x + 8)}^{2}}$ into the numerator.

$\frac{- 12}{{(x + 8)}^{2}} \cdot \frac{(x + 8) (x + 10)}{x + 10}$

Step 10.1.2

Factor $x + 8$ out of ${(x + 8)}^{2}$ .

$\frac{- 12}{(x + 8) (x + 8)} \cdot \frac{(x + 8) (x + 10)}{x + 10}$

Step 10.1.3

Cancel the common factor.

$\frac{- 12}{(x + 8) (x + 8)} \cdot \frac{(x + 8) (x + 10)}{x + 10}$

Step 10.1.4

Rewrite the expression.

$\frac{- 12}{x + 8} \cdot \frac{x + 10}{x + 10}$

Step 10.2

Multiply $\frac{- 12}{x + 8}$ by $\frac{x + 10}{x + 10}$ .

$\frac{- 12 (x + 10)}{(x + 8) (x + 10)}$

Step 10.3

Cancel the common factor of $x + 10$ .

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Step 10.3.1

Cancel the common factor.

$\frac{- 12 (x + 10)}{(x + 8) (x + 10)}$

Step 10.3.2

Rewrite the expression.

$\frac{- 12}{x + 8}$

Step 10.4

Move the negative in front of the fraction.

$- \frac{12}{x + 8}$