Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $x^{2}$ .

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Step 1.1

Multiply $\frac{\frac{1}{x^{2}} - 1}{3 + \frac{3}{x}}$ by $\frac{x^{2}}{x^{2}}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x^{2}$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Move $- 1$ to the left of $x^{2}$ .

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

Step 4.2

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

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

Step 4.3

Rewrite $1 - x^{2}$ in a factored form.

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Step 4.3.1

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

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

Step 4.3.2

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 $b = x$ .

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

Factor $x \cdot 3$ out of $x \cdot 3 (x) + x \cdot 3 (1)$ .

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

Step 5.2

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

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Step 5.2.1

Reorder terms.

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{1 - x}{x \cdot 3}$

Step 5.3

Move $3$ to the left of $x$ .

$\frac{1 - x}{3 x}$