Algebra Examples

$\frac{1}{x + 4} + \frac{8}{x^{2} - 16} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{1}{x + 4} + \frac{8}{x^{2} - 16} - 1 = 0$

Step 2

Simplify $\frac{1}{x + 4} + \frac{8}{x^{2} - 16} - 1$ .

Tap for more steps...

Step 2.1

Simplify the denominator.

Tap for more steps...

Step 2.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{1}{x + 4} + \frac{8}{x^{2} - 4^{2}} - 1 = 0$

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 $b = 4$ .

$\frac{1}{x + 4} + \frac{8}{(x + 4) (x - 4)} - 1 = 0$

Step 2.2

To write $\frac{1}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{1}{x + 4} \cdot \frac{x - 4}{x - 4} + \frac{8}{(x + 4) (x - 4)} - 1 = 0$

Step 2.3

Multiply $\frac{1}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{x - 4}{(x + 4) (x - 4)} + \frac{8}{(x + 4) (x - 4)} - 1 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x - 4 + 8}{(x + 4) (x - 4)} - 1 = 0$

Step 2.5

Add $- 4$ and $8$ .

$\frac{x + 4}{(x + 4) (x - 4)} - 1 = 0$

Step 2.6

Cancel the common factor of $x + 4$ .

Tap for more steps...

Step 2.6.1

Cancel the common factor.

$\frac{x + 4}{(x + 4) (x - 4)} - 1 = 0$

Step 2.6.2

Rewrite the expression.

$\frac{1}{x - 4} - 1 = 0$

Step 2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{1}{x - 4} - 1 \cdot \frac{x - 4}{x - 4} = 0$

Step 2.8

Combine $- 1$ and $\frac{x - 4}{x - 4}$ .

$\frac{1}{x - 4} + \frac{- (x - 4)}{x - 4} = 0$

Step 2.9

Combine the numerators over the common denominator.

$\frac{1 - (x - 4)}{x - 4} = 0$

Step 2.10

Simplify the numerator.

Tap for more steps...

Step 2.10.1

Apply the distributive property.

$\frac{1 - x - - 4}{x - 4} = 0$

Step 2.10.2

Multiply $- 1$ by $- 4$ .

$\frac{1 - x + 4}{x - 4} = 0$

Step 2.10.3

Add $1$ and $4$ .

$\frac{- x + 5}{x - 4} = 0$

Step 2.11

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

$\frac{- (x) + 5}{x - 4} = 0$

Step 2.12

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{- (x) - 1 (- 5)}{x - 4} = 0$

Step 2.13

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$\frac{- (x - 5)}{x - 4} = 0$

Step 2.14

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{- 1 (x - 5)}{x - 4} = 0$

Step 2.15

Move the negative in front of the fraction.

$- \frac{x - 5}{x - 4} = 0$

Step 3

Set the numerator equal to zero.

$x - 5 = 0$

Step 4

Add $5$ to both sides of the equation.

$x = 5$