Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Factor $3$ out of $- 3$ .

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

Step 2.1.3

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

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

Step 2.2

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

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

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 $b = 1$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Factor $x$ out of $3 x$ .

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

Step 3.1.3

Factor $x$ out of $x \cdot x + x \cdot 3$ .

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

Step 3.2

Cancel the common factor of $x + 1$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x (x + 3)$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$3 (x - 1)$

Step 3.4

Apply the distributive property.

$3 x + 3 \cdot - 1$

Step 3.5

Multiply $3$ by $- 1$ .

$3 x - 3$