Algebra Examples

$\frac{2}{x^{2} - 36} - \frac{1}{x^{2} + 6 x}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{2}{x^{2} - 6^{2}} - \frac{1}{x^{2} + 6 x}$

Step 1.1.2

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 $b = 6$ .

$\frac{2}{(x + 6) (x - 6)} - \frac{1}{x^{2} + 6 x}$

Step 1.2

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

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Step 1.2.1

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

$\frac{2}{(x + 6) (x - 6)} - \frac{1}{x \cdot x + 6 x}$

Step 1.2.2

Factor $x$ out of $6 x$ .

$\frac{2}{(x + 6) (x - 6)} - \frac{1}{x \cdot x + x \cdot 6}$

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot 6$ .

$\frac{2}{(x + 6) (x - 6)} - \frac{1}{x (x + 6)}$

Step 2

To write $\frac{2}{(x + 6) (x - 6)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2}{(x + 6) (x - 6)} \cdot \frac{x}{x} - \frac{1}{x (x + 6)}$

Step 3

To write $- \frac{1}{x (x + 6)}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{2}{(x + 6) (x - 6)} \cdot \frac{x}{x} - \frac{1}{x (x + 6)} \cdot \frac{x - 6}{x - 6}$

Step 4

Write each expression with a common denominator of $(x + 6) (x - 6) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{(x + 6) (x - 6)}$ by $\frac{x}{x}$ .

$\frac{2 x}{(x + 6) (x - 6) x} - \frac{1}{x (x + 6)} \cdot \frac{x - 6}{x - 6}$

Step 4.2

Multiply $\frac{1}{x (x + 6)}$ by $\frac{x - 6}{x - 6}$ .

$\frac{2 x}{(x + 6) (x - 6) x} - \frac{x - 6}{x (x + 6) (x - 6)}$

Step 4.3

Reorder the factors of $(x + 6) (x - 6) x$ .

$\frac{2 x}{x (x + 6) (x - 6)} - \frac{x - 6}{x (x + 6) (x - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 x - (x - 6)}{x (x + 6) (x - 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 x - x - - 6}{x (x + 6) (x - 6)}$

Step 6.2

Multiply $- 1$ by $- 6$ .

$\frac{2 x - x + 6}{x (x + 6) (x - 6)}$

Step 6.3

Subtract $x$ from $2 x$ .

$\frac{x + 6}{x (x + 6) (x - 6)}$

Step 7

Cancel the common factor of $x + 6$ .

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Step 7.1

Cancel the common factor.

$\frac{x + 6}{x (x + 6) (x - 6)}$

Step 7.2

Rewrite the expression.

$\frac{1}{x (x - 6)}$