Algebra Examples

$\frac{16}{x^{2} - 16} - \frac{2}{x - 4} = \frac{3}{x + 4}$

Step 1

Subtract $\frac{3}{x + 4}$ from both sides of the equation.

$\frac{16}{x^{2} - 16} - \frac{2}{x - 4} - \frac{3}{x + 4} = 0$

Step 2

Simplify $\frac{16}{x^{2} - 16} - \frac{2}{x - 4} - \frac{3}{x + 4}$ .

Tap for more steps...

Step 2.1

Simplify the denominator.

Tap for more steps...

Step 2.1.1

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

$\frac{16}{x^{2} - 4^{2}} - \frac{2}{x - 4} - \frac{3}{x + 4} = 0$

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 $b = 4$ .

$\frac{16}{(x + 4) (x - 4)} - \frac{2}{x - 4} - \frac{3}{x + 4} = 0$

Step 2.2

Find the common denominator.

Tap for more steps...

Step 2.2.1

Multiply $\frac{2}{x - 4}$ by $\frac{x + 4}{x + 4}$ .

$\frac{16}{(x + 4) (x - 4)} - (\frac{2}{x - 4} \cdot \frac{x + 4}{x + 4}) - \frac{3}{x + 4} = 0$

Step 2.2.2

Multiply $\frac{2}{x - 4}$ by $\frac{x + 4}{x + 4}$ .

$\frac{16}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x - 4) (x + 4)} - \frac{3}{x + 4} = 0$

Step 2.2.3

Multiply $\frac{3}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{16}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x - 4) (x + 4)} - (\frac{3}{x + 4} \cdot \frac{x - 4}{x - 4}) = 0$

Step 2.2.4

Multiply $\frac{3}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{16}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x - 4) (x + 4)} - \frac{3 (x - 4)}{(x + 4) (x - 4)} = 0$

Step 2.2.5

Reorder the factors of $(x - 4) (x + 4)$ .

$\frac{16}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x + 4) (x - 4)} - \frac{3 (x - 4)}{(x + 4) (x - 4)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{16 - 2 (x + 4) - 3 (x - 4)}{(x + 4) (x - 4)} = 0$

Step 2.4

Simplify each term.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$\frac{16 - 2 x - 2 \cdot 4 - 3 (x - 4)}{(x + 4) (x - 4)} = 0$

Step 2.4.2

Multiply $- 2$ by $4$ .

$\frac{16 - 2 x - 8 - 3 (x - 4)}{(x + 4) (x - 4)} = 0$

Step 2.4.3

Apply the distributive property.

$\frac{16 - 2 x - 8 - 3 x - 3 \cdot - 4}{(x + 4) (x - 4)} = 0$

Step 2.4.4

Multiply $- 3$ by $- 4$ .

$\frac{16 - 2 x - 8 - 3 x + 12}{(x + 4) (x - 4)} = 0$

Step 2.5

Subtract $8$ from $16$ .

$\frac{- 2 x + 8 - 3 x + 12}{(x + 4) (x - 4)} = 0$

Step 2.6

Subtract $3 x$ from $- 2 x$ .

$\frac{- 5 x + 8 + 12}{(x + 4) (x - 4)} = 0$

Step 2.7

Add $8$ and $12$ .

$\frac{- 5 x + 20}{(x + 4) (x - 4)} = 0$

Step 2.8

Factor $5$ out of $- 5 x + 20$ .

Tap for more steps...

Step 2.8.1

Factor $5$ out of $- 5 x$ .

$\frac{5 (- x) + 20}{(x + 4) (x - 4)} = 0$

Step 2.8.2

Factor $5$ out of $20$ .

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

Step 2.8.3

Factor $5$ out of $5 (- x) + 5 (4)$ .

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

Step 2.9

Cancel the common factor of $- x + 4$ and $x - 4$ .

Tap for more steps...

Step 2.9.1

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

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

Step 2.9.2

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 2.9.3

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

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

Step 2.9.4

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

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

Step 2.9.5

Cancel the common factor.

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

Step 2.9.6

Rewrite the expression.

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

Step 2.10

Multiply $5$ by $- 1$ .

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

Step 2.11

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$5 = 0$

Step 4

Since $5 \neq 0$ , there are no solutions.

No solution