Algebra Examples

$\frac{6 x^{2} - 54 x + 84}{8 x^{2} - 40 x + 48} \div \frac{x^{2} + x - 56}{2 x^{2} + 12 x - 32}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 x^{2} - 54 x + 84}{8 x^{2} - 40 x + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2

Cancel the common factor of $6 x^{2} - 54 x + 84$ and $8 x^{2} - 40 x + 48$ .

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Step 2.1

Factor $2$ out of $6 x^{2}$ .

$\frac{2 (3 x^{2}) - 54 x + 84}{8 x^{2} - 40 x + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.2

Factor $2$ out of $- 54 x$ .

$\frac{2 (3 x^{2}) + 2 (- 27 x) + 84}{8 x^{2} - 40 x + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.3

Factor $2$ out of $2 (3 x^{2}) + 2 (- 27 x)$ .

$\frac{2 (3 x^{2} - 27 x) + 84}{8 x^{2} - 40 x + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.4

Factor $2$ out of $84$ .

$\frac{2 (3 x^{2} - 27 x) + 2 (42)}{8 x^{2} - 40 x + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.5

Factor $2$ out of $2 (3 x^{2} - 27 x) + 2 (42)$ .

$\frac{2 (3 x^{2} - 27 x + 42)}{8 x^{2} - 40 x + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.6

Cancel the common factors.

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Step 2.6.1

Factor $2$ out of $8 x^{2}$ .

$\frac{2 (3 x^{2} - 27 x + 42)}{2 (4 x^{2}) - 40 x + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.6.2

Factor $2$ out of $- 40 x$ .

$\frac{2 (3 x^{2} - 27 x + 42)}{2 (4 x^{2}) + 2 (- 20 x) + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.6.3

Factor $2$ out of $2 (4 x^{2}) + 2 (- 20 x)$ .

$\frac{2 (3 x^{2} - 27 x + 42)}{2 (4 x^{2} - 20 x) + 48} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.6.4

Factor $2$ out of $48$ .

$\frac{2 (3 x^{2} - 27 x + 42)}{2 (4 x^{2} - 20 x) + 2 (24)} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.6.5

Factor $2$ out of $2 (4 x^{2} - 20 x) + 2 (24)$ .

$\frac{2 (3 x^{2} - 27 x + 42)}{2 (4 x^{2} - 20 x + 24)} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.6.6

Cancel the common factor.

$\frac{2 (3 x^{2} - 27 x + 42)}{2 (4 x^{2} - 20 x + 24)} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 2.6.7

Rewrite the expression.

$\frac{3 x^{2} - 27 x + 42}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $3$ out of $3 x^{2} - 27 x + 42$ .

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Step 3.1.1

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

$\frac{3 (x^{2}) - 27 x + 42}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 3.1.2

Factor $3$ out of $- 27 x$ .

$\frac{3 (x^{2}) + 3 (- 9 x) + 42}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 3.1.3

Factor $3$ out of $42$ .

$\frac{3 x^{2} + 3 (- 9 x) + 3 \cdot 14}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 3.1.4

Factor $3$ out of $3 x^{2} + 3 (- 9 x)$ .

$\frac{3 (x^{2} - 9 x) + 3 \cdot 14}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 3.1.5

Factor $3$ out of $3 (x^{2} - 9 x) + 3 \cdot 14$ .

$\frac{3 (x^{2} - 9 x + 14)}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 3.2

Factor $x^{2} - 9 x + 14$ using the AC method.

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Step 3.2.1

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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 3.2.2

Write the factored form using these integers.

$\frac{3 ((x - 7) (x - 2))}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

$\frac{3 (x - 7) (x - 2)}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $4$ out of $4 x^{2} - 20 x + 24$ .

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Step 4.1.1

Factor $4$ out of $4 x^{2}$ .

$\frac{3 (x - 7) (x - 2)}{4 (x^{2}) - 20 x + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 4.1.2

Factor $4$ out of $- 20 x$ .

$\frac{3 (x - 7) (x - 2)}{4 (x^{2}) + 4 (- 5 x) + 24} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 4.1.3

Factor $4$ out of $24$ .

$\frac{3 (x - 7) (x - 2)}{4 x^{2} + 4 (- 5 x) + 4 \cdot 6} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 4.1.4

Factor $4$ out of $4 x^{2} + 4 (- 5 x)$ .

$\frac{3 (x - 7) (x - 2)}{4 (x^{2} - 5 x) + 4 \cdot 6} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 4.1.5

Factor $4$ out of $4 (x^{2} - 5 x) + 4 \cdot 6$ .

$\frac{3 (x - 7) (x - 2)}{4 (x^{2} - 5 x + 6)} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 4.2

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 4.2.1

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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 4.2.2

Write the factored form using these integers.

$\frac{3 (x - 7) (x - 2)}{4 ((x - 3) (x - 2))} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

$\frac{3 (x - 7) (x - 2)}{4 (x - 3) (x - 2)} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 5

Cancel the common factor of $x - 2$ .

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Step 5.1

Cancel the common factor.

$\frac{3 (x - 7) (x - 2)}{4 (x - 3) (x - 2)} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 5.2

Rewrite the expression.

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 x^{2} + 12 x - 32}{x^{2} + x - 56}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $2 x^{2} + 12 x - 32$ .

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Step 6.1.1

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

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 (x^{2}) + 12 x - 32}{x^{2} + x - 56}$

Step 6.1.2

Factor $2$ out of $12 x$ .

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 (x^{2}) + 2 (6 x) - 32}{x^{2} + x - 56}$

Step 6.1.3

Factor $2$ out of $- 32$ .

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 x^{2} + 2 (6 x) + 2 \cdot - 16}{x^{2} + x - 56}$

Step 6.1.4

Factor $2$ out of $2 x^{2} + 2 (6 x)$ .

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 (x^{2} + 6 x) + 2 \cdot - 16}{x^{2} + x - 56}$

Step 6.1.5

Factor $2$ out of $2 (x^{2} + 6 x) + 2 \cdot - 16$ .

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 (x^{2} + 6 x - 16)}{x^{2} + x - 56}$

Step 6.2

Factor $x^{2} + 6 x - 16$ using the AC method.

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Step 6.2.1

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 $- 16$ and whose sum is $6$ .

$- 2, 8$

Step 6.2.2

Write the factored form using these integers.

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 ((x - 2) (x + 8))}{x^{2} + x - 56}$

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 (x - 2) (x + 8)}{x^{2} + x - 56}$

Step 7

Factor $x^{2} + x - 56$ 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 $- 56$ and whose sum is $1$ .

$- 7, 8$

Step 7.2

Write the factored form using these integers.

$\frac{3 (x - 7)}{4 (x - 3)} \cdot \frac{2 (x - 2) (x + 8)}{(x - 7) (x + 8)}$

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of $x - 7$ .

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Step 8.1.1

Factor $x - 7$ out of $3 (x - 7)$ .

$\frac{(x - 7) \cdot 3}{4 (x - 3)} \cdot \frac{2 (x - 2) (x + 8)}{(x - 7) (x + 8)}$

Step 8.1.2

Cancel the common factor.

$\frac{(x - 7) \cdot 3}{4 (x - 3)} \cdot \frac{2 (x - 2) (x + 8)}{(x - 7) (x + 8)}$

Step 8.1.3

Rewrite the expression.

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

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Factor $2$ out of $4 (x - 3)$ .

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

Step 8.2.2

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

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

Step 8.2.3

Cancel the common factor.

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

Step 8.2.4

Rewrite the expression.

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

Step 8.3

Multiply $\frac{3}{2 (x - 3)}$ by $\frac{(x - 2) (x + 8)}{x + 8}$ .

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

Step 8.4

Cancel the common factor of $x + 8$ .

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Step 8.4.1

Cancel the common factor.

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

Step 8.4.2

Rewrite the expression.

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