Algebra Examples

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

Step 1

Subtract $\frac{16}{x^{2} - 64}$ from both sides of the equation.

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

Step 2

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

Tap for more steps...

Step 2.1

Simplify the denominator.

Tap for more steps...

Step 2.1.1

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

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

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 $b = 8$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Write each expression with a common denominator of $(x - 8) (x + 8)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Reorder the factors of $(x - 8) (x + 8)$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Combine the opposite terms in $\frac{x + 8 + x - 8}{(x + 8) (x - 8)} - \frac{16}{(x + 8) (x - 8)}$ .

Tap for more steps...

Step 2.6.1

Subtract $8$ from $8$ .

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

Step 2.6.2

Add $x$ and $0$ .

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Add $x$ and $x$ .

$\frac{2 x - 16}{(x + 8) (x - 8)} = 0$

Step 2.9

Factor $2$ out of $2 x - 16$ .

Tap for more steps...

Step 2.9.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 16}{(x + 8) (x - 8)} = 0$

Step 2.9.2

Factor $2$ out of $- 16$ .

$\frac{2 x + 2 \cdot - 8}{(x + 8) (x - 8)} = 0$

Step 2.9.3

Factor $2$ out of $2 x + 2 \cdot - 8$ .

$\frac{2 (x - 8)}{(x + 8) (x - 8)} = 0$

Step 2.10

Cancel the common factor of $x - 8$ .

Tap for more steps...

Step 2.10.1

Cancel the common factor.

$\frac{2 (x - 8)}{(x + 8) (x - 8)} = 0$

Step 2.10.2

Rewrite the expression.

$\frac{2}{x + 8} = 0$

Step 3

Set the numerator equal to zero.

$2 = 0$

Step 4

Since $2 \neq 0$ , there are no solutions.

No solution