Algebra Examples

$\frac{2 - 18 x^{2}}{3 x^{2} - 8 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2$ out of $2 - 18 x^{2}$ .

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Step 1.1.1

Factor $2$ out of $2$ .

$\frac{2 (1) - 18 x^{2}}{3 x^{2} - 8 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 1.1.2

Factor $2$ out of $- 18 x^{2}$ .

$\frac{2 (1) + 2 (- 9 x^{2})}{3 x^{2} - 8 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 1.1.3

Factor $2$ out of $2 (1) + 2 (- 9 x^{2})$ .

$\frac{2 (1 - 9 x^{2})}{3 x^{2} - 8 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 1.2

Rewrite $1$ as $1^{2}$ .

$\frac{2 (1^{2} - 9 x^{2})}{3 x^{2} - 8 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 1.3

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{2 (1^{2} - {(3 x)}^{2})}{3 x^{2} - 8 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 3 x$ .

$\frac{2 ((1 + 3 x) (1 - (3 x)))}{3 x^{2} - 8 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 1.5

Multiply $3$ by $- 1$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{3 x^{2} - 8 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = - 8$ .

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Step 2.1.1

Factor $- 8$ out of $- 8 x$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{3 x^{2} - 8 (x) - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 2.1.2

Rewrite $- 8$ as $1$ plus $- 9$

$\frac{2 (1 + 3 x) (1 - 3 x)}{3 x^{2} + (1 - 9) x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 2.1.3

Apply the distributive property.

$\frac{2 (1 + 3 x) (1 - 3 x)}{3 x^{2} + 1 x - 9 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 2.1.4

Multiply $x$ by $1$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{3 x^{2} + x - 9 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x^{2} + x) - 9 x - 3} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (1 + 3 x) (1 - 3 x)}{x (3 x + 1) - 3 (3 x + 1)} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{3 x^{2} - 7 x - 6}{12 x^{2} + 80 x - 28}$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 6 = - 18$ and whose sum is $b = - 7$ .

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Step 3.1.1

Factor $- 7$ out of $- 7 x$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{3 x^{2} - 7 (x) - 6}{12 x^{2} + 80 x - 28}$

Step 3.1.2

Rewrite $- 7$ as $2$ plus $- 9$

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{3 x^{2} + (2 - 9) x - 6}{12 x^{2} + 80 x - 28}$

Step 3.1.3

Apply the distributive property.

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{3 x^{2} + 2 x - 9 x - 6}{12 x^{2} + 80 x - 28}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x^{2} + 2 x) - 9 x - 6}{12 x^{2} + 80 x - 28}$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{x (3 x + 2) - 3 (3 x + 2)}{12 x^{2} + 80 x - 28}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{12 x^{2} + 80 x - 28}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $4$ out of $12 x^{2} + 80 x - 28$ .

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Step 4.1.1

Factor $4$ out of $12 x^{2}$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x^{2}) + 80 x - 28}$

Step 4.1.2

Factor $4$ out of $80 x$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x^{2}) + 4 (20 x) - 28}$

Step 4.1.3

Factor $4$ out of $- 28$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x^{2}) + 4 (20 x) + 4 (- 7)}$

Step 4.1.4

Factor $4$ out of $4 (3 x^{2}) + 4 (20 x)$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x^{2} + 20 x) + 4 (- 7)}$

Step 4.1.5

Factor $4$ out of $4 (3 x^{2} + 20 x) + 4 (- 7)$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x^{2} + 20 x - 7)}$

Step 4.2

Factor by grouping.

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Step 4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 7 = - 21$ and whose sum is $b = 20$ .

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Step 4.2.1.1

Factor $20$ out of $20 x$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x^{2} + 20 (x) - 7)}$

Step 4.2.1.2

Rewrite $20$ as $- 1$ plus $21$

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x^{2} + (- 1 + 21) x - 7)}$

Step 4.2.1.3

Apply the distributive property.

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x^{2} - 1 x + 21 x - 7)}$

Step 4.2.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1

Group the first two terms and the last two terms.

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 ((3 x^{2} - 1 x) + 21 x - 7)}$

Step 4.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (x (3 x - 1) + 7 (3 x - 1))}$

Step 4.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 ((3 x - 1) (x + 7))}$

$\frac{2 (1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x - 1) (x + 7)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $2$ .

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Step 5.1.1

Factor $2$ out of $2 (1 + 3 x) (1 - 3 x)$ .

$\frac{2 ((1 + 3 x) (1 - 3 x))}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{4 (3 x - 1) (x + 7)}$

Step 5.1.2

Factor $2$ out of $4 (3 x - 1) (x + 7)$ .

$\frac{2 ((1 + 3 x) (1 - 3 x))}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{2 (2 (3 x - 1) (x + 7))}$

Step 5.1.3

Cancel the common factor.

$\frac{2 ((1 + 3 x) (1 - 3 x))}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{2 (2 (3 x - 1) (x + 7))}$

Step 5.1.4

Rewrite the expression.

$\frac{(1 + 3 x) (1 - 3 x)}{(3 x + 1) (x - 3)} \cdot \frac{(3 x + 2) (x - 3)}{2 (3 x - 1) (x + 7)}$

Step 5.2

Cancel the common factor of $x - 3$ .

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Step 5.2.1

Factor $x - 3$ out of $(3 x + 1) (x - 3)$ .

$\frac{(1 + 3 x) (1 - 3 x)}{(x - 3) (3 x + 1)} \cdot \frac{(3 x + 2) (x - 3)}{2 (3 x - 1) (x + 7)}$

Step 5.2.2

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

$\frac{(1 + 3 x) (1 - 3 x)}{(x - 3) (3 x + 1)} \cdot \frac{(x - 3) (3 x + 2)}{2 (3 x - 1) (x + 7)}$

Step 5.2.3

Cancel the common factor.

$\frac{(1 + 3 x) (1 - 3 x)}{(x - 3) (3 x + 1)} \cdot \frac{(x - 3) (3 x + 2)}{2 (3 x - 1) (x + 7)}$

Step 5.2.4

Rewrite the expression.

$\frac{(1 + 3 x) (1 - 3 x)}{3 x + 1} \cdot \frac{3 x + 2}{2 (3 x - 1) (x + 7)}$

Step 5.3

Multiply $\frac{(1 + 3 x) (1 - 3 x)}{3 x + 1}$ by $\frac{3 x + 2}{2 (3 x - 1) (x + 7)}$ .

$\frac{(1 + 3 x) (1 - 3 x) (3 x + 2)}{(3 x + 1) (2 (3 x - 1) (x + 7))}$

Step 5.4

Cancel the common factor of $1 + 3 x$ and $3 x + 1$ .

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Step 5.4.1

Reorder terms.

$\frac{(3 x + 1) (1 - 3 x) (3 x + 2)}{(3 x + 1) (2 (3 x - 1) (x + 7))}$

Step 5.4.2

Cancel the common factor.

$\frac{(3 x + 1) (1 - 3 x) (3 x + 2)}{(3 x + 1) (2 (3 x - 1) (x + 7))}$

Step 5.4.3

Rewrite the expression.

$\frac{(1 - 3 x) (3 x + 2)}{2 (3 x - 1) (x + 7)}$

Step 5.5

Cancel the common factor of $1 - 3 x$ and $3 x - 1$ .

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Step 5.5.1

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

$\frac{(- 1 (- 1) - 3 x) (3 x + 2)}{2 (3 x - 1) (x + 7)}$

Step 5.5.2

Factor $- 1$ out of $- 3 x$ .

$\frac{(- 1 (- 1) - (3 x)) (3 x + 2)}{2 (3 x - 1) (x + 7)}$

Step 5.5.3

Factor $- 1$ out of $- 1 (- 1) - (3 x)$ .

$\frac{- 1 (- 1 + 3 x) (3 x + 2)}{2 (3 x - 1) (x + 7)}$

Step 5.5.4

Reorder terms.

$\frac{- 1 (3 x - 1) (3 x + 2)}{2 (3 x - 1) (x + 7)}$

Step 5.5.5

Cancel the common factor.

$\frac{- 1 (3 x - 1) (3 x + 2)}{2 (3 x - 1) (x + 7)}$

Step 5.5.6

Rewrite the expression.

$\frac{- 1 (3 x + 2)}{2 (x + 7)}$

Step 5.6

Move the negative in front of the fraction.

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