Solve by Factoring 1/(x+6)+12/(x^2-36)=1

$\frac{1}{x + 6} + \frac{12}{x^{2} - 36} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{1}{x + 6} + \frac{12}{x^{2} - 36} - 1 = 0$

Step 2

Simplify $\frac{1}{x + 6} + \frac{12}{x^{2} - 36} - 1$ .

Simplify the denominator.

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

$\frac{1}{x + 6} + \frac{12}{x^{2} - 6^{2}} - 1 = 0$

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 $b = 6$ .

$\frac{1}{x + 6} + \frac{12}{(x + 6) (x - 6)} - 1 = 0$

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

$\frac{1}{x + 6} \cdot \frac{x - 6}{x - 6} + \frac{12}{(x + 6) (x - 6)} - 1 = 0$

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

$\frac{x - 6}{(x + 6) (x - 6)} + \frac{12}{(x + 6) (x - 6)} - 1 = 0$

Combine the numerators over the common denominator.

$\frac{x - 6 + 12}{(x + 6) (x - 6)} - 1 = 0$

Add $- 6$ and $12$ .

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

Cancel the common factor of $x + 6$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x - 6} - 1 = 0$

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

$\frac{1}{x - 6} - 1 \cdot \frac{x - 6}{x - 6} = 0$

Combine $- 1$ and $\frac{x - 6}{x - 6}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

$\frac{1 - x - - 6}{x - 6} = 0$

Multiply $- 1$ by $- 6$ .

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

Add $1$ and $6$ .

$\frac{- x + 7}{x - 6} = 0$

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 7}{x - 6} = 0$

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{- (x) - 1 (- 7)}{x - 6} = 0$

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

$\frac{- (x - 7)}{x - 6} = 0$

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$\frac{- 1 (x - 7)}{x - 6} = 0$

Move the negative in front of the fraction.

$- \frac{x - 7}{x - 6} = 0$

Step 3

Set the numerator equal to zero.

$x - 7 = 0$

Step 4

Add $7$ to both sides of the equation.

$x = 7$

Solve by Factoring 1/(x+6)+12/(x^2-36)=1

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