Solve by Factoring 1/(x+8)+16/(x^2-64)=1

$\frac{1}{x + 8} + \frac{16}{x^{2} - 64} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{1}{x + 8} + \frac{16}{x^{2} - 64} - 1 = 0$

Step 2

Simplify $\frac{1}{x + 8} + \frac{16}{x^{2} - 64} - 1$ .

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$\frac{1}{x + 8} + \frac{16}{x^{2} - 8^{2}} - 1 = 0$

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 $b = 8$ .

$\frac{1}{x + 8} + \frac{16}{(x + 8) (x - 8)} - 1 = 0$

To write $\frac{1}{x + 8}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{1}{x + 8} \cdot \frac{x - 8}{x - 8} + \frac{16}{(x + 8) (x - 8)} - 1 = 0$

Multiply $\frac{1}{x + 8}$ by $\frac{x - 8}{x - 8}$ .

$\frac{x - 8}{(x + 8) (x - 8)} + \frac{16}{(x + 8) (x - 8)} - 1 = 0$

Combine the numerators over the common denominator.

$\frac{x - 8 + 16}{(x + 8) (x - 8)} - 1 = 0$

Add $- 8$ and $16$ .

$\frac{x + 8}{(x + 8) (x - 8)} - 1 = 0$

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$\frac{x + 8}{(x + 8) (x - 8)} - 1 = 0$

Rewrite the expression.

$\frac{1}{x - 8} - 1 = 0$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{1}{x - 8} - 1 \cdot \frac{x - 8}{x - 8} = 0$

Combine $- 1$ and $\frac{x - 8}{x - 8}$ .

$\frac{1}{x - 8} + \frac{- (x - 8)}{x - 8} = 0$

Combine the numerators over the common denominator.

$\frac{1 - (x - 8)}{x - 8} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{1 - x - - 8}{x - 8} = 0$

Multiply $- 1$ by $- 8$ .

$\frac{1 - x + 8}{x - 8} = 0$

Add $1$ and $8$ .

$\frac{- x + 9}{x - 8} = 0$

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

$\frac{- (x) + 9}{x - 8} = 0$

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (x) - 1 (- 9)}{x - 8} = 0$

Factor $- 1$ out of $- (x) - 1 (- 9)$ .

$\frac{- (x - 9)}{x - 8} = 0$

Rewrite $- (x - 9)$ as $- 1 (x - 9)$ .

$\frac{- 1 (x - 9)}{x - 8} = 0$

Move the negative in front of the fraction.

$- \frac{x - 9}{x - 8} = 0$

Step 3

Set the numerator equal to zero.

$x - 9 = 0$

Step 4

Add $9$ to both sides of the equation.

$x = 9$

Solve by Factoring 1/(x+8)+16/(x^2-64)=1

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