Algebra Examples

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

Step 1

Subtract $\frac{1}{x + 6}$ from both sides of the equation.

$\frac{x}{x^{2} - 36} + \frac{2}{x - 6} - \frac{1}{x + 6} = 0$

Step 2

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

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$\frac{x}{x^{2} - 6^{2}} + \frac{2}{x - 6} - \frac{1}{x + 6} = 0$

Step 2.1.2

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 $b = 6$ .

$\frac{x}{(x + 6) (x - 6)} + \frac{2}{x - 6} - \frac{1}{x + 6} = 0$

Step 2.2

Find the common denominator.

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Step 2.2.1

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

$\frac{x}{(x + 6) (x - 6)} + \frac{2}{x - 6} \cdot \frac{x + 6}{x + 6} - \frac{1}{x + 6} = 0$

Step 2.2.2

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

$\frac{x}{(x + 6) (x - 6)} + \frac{2 (x + 6)}{(x - 6) (x + 6)} - \frac{1}{x + 6} = 0$

Step 2.2.3

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

$\frac{x}{(x + 6) (x - 6)} + \frac{2 (x + 6)}{(x - 6) (x + 6)} - (\frac{1}{x + 6} \cdot \frac{x - 6}{x - 6}) = 0$

Step 2.2.4

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

$\frac{x}{(x + 6) (x - 6)} + \frac{2 (x + 6)}{(x - 6) (x + 6)} - \frac{x - 6}{(x + 6) (x - 6)} = 0$

Step 2.2.5

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

$\frac{x}{(x + 6) (x - 6)} + \frac{2 (x + 6)}{(x + 6) (x - 6)} - \frac{x - 6}{(x + 6) (x - 6)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x + 2 (x + 6) - (x - 6)}{(x + 6) (x - 6)} = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

Apply the distributive property.

$\frac{x + 2 x + 2 \cdot 6 - (x - 6)}{(x + 6) (x - 6)} = 0$

Step 2.4.2

Multiply $2$ by $6$ .

$\frac{x + 2 x + 12 - (x - 6)}{(x + 6) (x - 6)} = 0$

Step 2.4.3

Apply the distributive property.

$\frac{x + 2 x + 12 - x - - 6}{(x + 6) (x - 6)} = 0$

Step 2.4.4

Multiply $- 1$ by $- 6$ .

$\frac{x + 2 x + 12 - x + 6}{(x + 6) (x - 6)} = 0$

Step 2.5

Combine the opposite terms in $x + 2 x + 12 - x + 6$ .

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Step 2.5.1

Subtract $x$ from $x$ .

$\frac{2 x + 12 + 0 + 6}{(x + 6) (x - 6)} = 0$

Step 2.5.2

Add $2 x + 12$ and $0$ .

$\frac{2 x + 12 + 6}{(x + 6) (x - 6)} = 0$

Step 2.6

Add $12$ and $6$ .

$\frac{2 x + 18}{(x + 6) (x - 6)} = 0$

Step 2.7

Factor $2$ out of $2 x + 18$ .

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Step 2.7.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 18}{(x + 6) (x - 6)} = 0$

Step 2.7.2

Factor $2$ out of $18$ .

$\frac{2 x + 2 \cdot 9}{(x + 6) (x - 6)} = 0$

Step 2.7.3

Factor $2$ out of $2 x + 2 \cdot 9$ .

$\frac{2 (x + 9)}{(x + 6) (x - 6)} = 0$

Step 3

Set the numerator equal to zero.

$2 (x + 9) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $2 (x + 9) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (x + 9) = 0$ by $2$ .

$\frac{2 (x + 9)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (x + 9)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $x + 9$ by $1$ .

$x + 9 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$x + 9 = 0$

Step 4.2

Subtract $9$ from both sides of the equation.

$x = - 9$