Algebra Examples

$\frac{3}{3 - x} \cdot \frac{2 x^{2} + 18 x + 16}{- x - 1} \div \frac{x^{2} + 11 x + 24}{9 - x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3}{3 - x} \cdot \frac{2 x^{2} + 18 x + 16}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2$ out of $2 x^{2} + 18 x + 16$ .

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Step 2.1.1

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

$\frac{3}{3 - x} \cdot \frac{2 (x^{2}) + 18 x + 16}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 2.1.2

Factor $2$ out of $18 x$ .

$\frac{3}{3 - x} \cdot \frac{2 (x^{2}) + 2 (9 x) + 16}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 2.1.3

Factor $2$ out of $16$ .

$\frac{3}{3 - x} \cdot \frac{2 x^{2} + 2 (9 x) + 2 \cdot 8}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 2.1.4

Factor $2$ out of $2 x^{2} + 2 (9 x)$ .

$\frac{3}{3 - x} \cdot \frac{2 (x^{2} + 9 x) + 2 \cdot 8}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 2.1.5

Factor $2$ out of $2 (x^{2} + 9 x) + 2 \cdot 8$ .

$\frac{3}{3 - x} \cdot \frac{2 (x^{2} + 9 x + 8)}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 2.2

Factor $x^{2} + 9 x + 8$ using the AC method.

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Step 2.2.1

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 $8$ and whose sum is $9$ .

$1, 8$

Step 2.2.2

Write the factored form using these integers.

$\frac{3}{3 - x} \cdot \frac{2 ((x + 1) (x + 8))}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

$\frac{3}{3 - x} \cdot \frac{2 (x + 1) (x + 8)}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $x + 1$ and $- x - 1$ .

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Step 3.1.1

Factor $- 1$ out of $x$ .

$\frac{3}{3 - x} \cdot \frac{2 (- 1 (- x) + 1) (x + 8)}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 3.1.2

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

$\frac{3}{3 - x} \cdot \frac{2 (- 1 (- x) - 1 (- 1)) (x + 8)}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 3.1.3

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

$\frac{3}{3 - x} \cdot \frac{2 (- 1 (- x - 1)) (x + 8)}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 3.1.4

Cancel the common factor.

$\frac{3}{3 - x} \cdot \frac{2 (- 1 (- x - 1)) (x + 8)}{- x - 1} \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 3.1.5

Divide $2 \cdot (- 1) (x + 8)$ by $1$ .

$\frac{3}{3 - x} \cdot (2 \cdot (- 1) (x + 8)) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$2 \cdot - 1 \frac{3}{3 - x} (x + 8) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 3.2.2

Multiply $2$ by $- 1$ .

$- 2 \frac{3}{3 - x} (x + 8) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 4

Multiply $- 2 \frac{3}{3 - x}$ .

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Step 4.1

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

$\frac{- 2 \cdot 3}{3 - x} (x + 8) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 4.2

Multiply $- 2$ by $3$ .

$\frac{- 6}{3 - x} (x + 8) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 5

Move the negative in front of the fraction.

$- \frac{6}{3 - x} (x + 8) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 6

Apply the distributive property.

$(- \frac{6}{3 - x} x - \frac{6}{3 - x} \cdot 8) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 7

Combine $x$ and $\frac{6}{3 - x}$ .

$(- \frac{x \cdot 6}{3 - x} - \frac{6}{3 - x} \cdot 8) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 8

Multiply $- \frac{6}{3 - x} \cdot 8$ .

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Step 8.1

Multiply $8$ by $- 1$ .

$(- \frac{x \cdot 6}{3 - x} - 8 \frac{6}{3 - x}) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 8.2

Combine $- 8$ and $\frac{6}{3 - x}$ .

$(- \frac{x \cdot 6}{3 - x} + \frac{- 8 \cdot 6}{3 - x}) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 8.3

Multiply $- 8$ by $6$ .

$(- \frac{x \cdot 6}{3 - x} + \frac{- 48}{3 - x}) \frac{9 - x^{2}}{x^{2} + 11 x + 24}$

Step 9

Simplify the numerator.

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Step 9.1

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

$(- \frac{x \cdot 6}{3 - x} + \frac{- 48}{3 - x}) \frac{3^{2} - x^{2}}{x^{2} + 11 x + 24}$

Step 9.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$(- \frac{x \cdot 6}{3 - x} + \frac{- 48}{3 - x}) \frac{(3 + x) (3 - x)}{x^{2} + 11 x + 24}$

Step 10

Factor $x^{2} + 11 x + 24$ using the AC method.

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Step 10.1

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 $24$ and whose sum is $11$ .

$3, 8$

Step 10.2

Write the factored form using these integers.

$(- \frac{x \cdot 6}{3 - x} + \frac{- 48}{3 - x}) \frac{(3 + x) (3 - x)}{(x + 3) (x + 8)}$

Step 11

Simplify terms.

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Step 11.1

Combine the numerators over the common denominator.

$\frac{- x \cdot 6 - 48}{3 - x} \cdot \frac{(3 + x) (3 - x)}{(x + 3) (x + 8)}$

Step 11.2

Multiply $6$ by $- 1$ .

$\frac{- 6 x - 48}{3 - x} \cdot \frac{(3 + x) (3 - x)}{(x + 3) (x + 8)}$

Step 11.3

Factor $6$ out of $- 6 x - 48$ .

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Step 11.3.1

Factor $6$ out of $- 6 x$ .

$\frac{6 (- x) - 48}{3 - x} \cdot \frac{(3 + x) (3 - x)}{(x + 3) (x + 8)}$

Step 11.3.2

Factor $6$ out of $- 48$ .

$\frac{6 (- x) + 6 (- 8)}{3 - x} \cdot \frac{(3 + x) (3 - x)}{(x + 3) (x + 8)}$

Step 11.3.3

Factor $6$ out of $6 (- x) + 6 (- 8)$ .

$\frac{6 (- x - 8)}{3 - x} \cdot \frac{(3 + x) (3 - x)}{(x + 3) (x + 8)}$

Step 11.4

Cancel the common factor of $3 - x$ .

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Step 11.4.1

Factor $3 - x$ out of $(3 + x) (3 - x)$ .

$\frac{6 (- x - 8)}{3 - x} \cdot \frac{(3 - x) (3 + x)}{(x + 3) (x + 8)}$

Step 11.4.2

Cancel the common factor.

$\frac{6 (- x - 8)}{3 - x} \cdot \frac{(3 - x) (3 + x)}{(x + 3) (x + 8)}$

Step 11.4.3

Rewrite the expression.

$6 (- x - 8) \frac{3 + x}{(x + 3) (x + 8)}$

Step 11.5

Combine $\frac{3 + x}{(x + 3) (x + 8)}$ and $6$ .

$\frac{(3 + x) \cdot 6}{(x + 3) (x + 8)} (- x - 8)$

Step 11.6

Cancel the common factor of $3 + x$ and $x + 3$ .

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Step 11.6.1

Reorder terms.

$\frac{(x + 3) \cdot 6}{(x + 3) (x + 8)} (- x - 8)$

Step 11.6.2

Cancel the common factor.

$\frac{(x + 3) \cdot 6}{(x + 3) (x + 8)} (- x - 8)$

Step 11.6.3

Rewrite the expression.

$\frac{6}{x + 8} (- x - 8)$

Step 11.7

Multiply $\frac{6}{x + 8}$ by $- x - 8$ .

$\frac{6 (- x - 8)}{x + 8}$

Step 11.8

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

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Step 11.8.1

Factor $- 1$ out of $- x$ .

$\frac{6 (- (x) - 8)}{x + 8}$

Step 11.8.2

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

$\frac{6 (- (x) - 1 (8))}{x + 8}$

Step 11.8.3

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

$\frac{6 (- (x + 8))}{x + 8}$

Step 11.8.4

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

$\frac{6 (- 1 (x + 8))}{x + 8}$

Step 11.8.5

Cancel the common factor.

$\frac{6 (- 1 (x + 8))}{x + 8}$

Step 11.8.6

Divide $6 \cdot (- 1)$ by $1$ .

$6 \cdot (- 1)$

Step 11.9

Multiply $6$ by $- 1$ .

$- 6$