Algebra Examples

$\frac{\frac{6 x^{2} - 6}{8 x^{2} + 8 x}}{\frac{3 - 3 x}{4 x^{2} + 4 x}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 x^{2} - 6}{8 x^{2} + 8 x} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

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

$\frac{6 (x^{2}) - 6}{8 x^{2} + 8 x} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

Step 2.1.2

Factor $6$ out of $- 6$ .

$\frac{6 (x^{2}) + 6 (- 1)}{8 x^{2} + 8 x} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

Step 2.1.3

Factor $6$ out of $6 (x^{2}) + 6 (- 1)$ .

$\frac{6 (x^{2} - 1)}{8 x^{2} + 8 x} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

Step 2.2

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

$\frac{6 (x^{2} - 1^{2})}{8 x^{2} + 8 x} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

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 $b = 1$ .

$\frac{6 (x + 1) (x - 1)}{8 x^{2} + 8 x} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

$\frac{6 (x + 1) (x - 1)}{8 x (x) + 8 x} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

Step 3.1.2

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

$\frac{6 (x + 1) (x - 1)}{8 x (x) + 8 x (1)} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

Step 3.1.3

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

$\frac{6 (x + 1) (x - 1)}{8 x (x + 1)} \cdot \frac{4 x^{2} + 4 x}{3 - 3 x}$

Step 3.2

Cancel the common factor of $6$ and $8$ .

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Step 3.2.1

Factor $2$ out of $6 (x + 1) (x - 1)$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $2$ out of $8 x (x + 1)$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 1$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Factor $4 x$ out of $4 x$ .

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

Step 3.4.3

Factor $4 x$ out of $4 x (x) + 4 x (1)$ .

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

Step 3.5

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

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Step 3.5.1

Factor $3$ out of $3$ .

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.6

Cancel the common factor of $3$ .

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

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

Step 3.7

Cancel the common factor of $4 x$ .

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Step 3.7.1

Cancel the common factor.

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

Step 3.7.2

Rewrite the expression.

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

Step 3.8

Multiply $x - 1$ by $\frac{x + 1}{1 - x}$ .

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

Step 3.9

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

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Step 3.9.1

Factor $- 1$ out of $x$ .

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

Step 3.9.2

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

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

Step 3.9.3

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

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

Step 3.9.4

Reorder terms.

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

Step 3.9.5

Cancel the common factor.

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

Step 3.9.6

Divide $- 1 (x + 1)$ by $1$ .

$- 1 (x + 1)$

Step 3.10

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

$- (x + 1)$

Step 3.11

Apply the distributive property.

$- x - 1 \cdot 1$

Step 3.12

Multiply $- 1$ by $1$ .

$- x - 1$