Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 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 $b = 6$ .

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.1

Move $x$ .

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

Step 4.2.2

Multiply $x$ by $x$ .

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

Step 4.3

Multiply $- 6$ by $- 6$ .

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

Step 4.4

Apply the distributive property.

$\frac{6 x^{2} - 6 x^{2} + 36 x + 6 x + 6 \cdot 6}{(x + 6) (x - 6)}$

Step 4.5

Multiply $6$ by $6$ .

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

Step 5

Simplify terms.

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Step 5.1

Subtract $6 x^{2}$ from $6 x^{2}$ .

$\frac{0 + 36 x + 6 x + 36}{(x + 6) (x - 6)}$

Step 5.2

Add $0$ and $36 x$ .

$\frac{36 x + 6 x + 36}{(x + 6) (x - 6)}$

Step 5.3

Add $36 x$ and $6 x$ .

$\frac{42 x + 36}{(x + 6) (x - 6)}$

Step 5.4

Factor $6$ out of $42 x + 36$ .

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Step 5.4.1

Factor $6$ out of $42 x$ .

$\frac{6 (7 x) + 36}{(x + 6) (x - 6)}$

Step 5.4.2

Factor $6$ out of $36$ .

$\frac{6 (7 x) + 6 (6)}{(x + 6) (x - 6)}$

Step 5.4.3

Factor $6$ out of $6 (7 x) + 6 (6)$ .

$\frac{6 (7 x + 6)}{(x + 6) (x - 6)}$