Trigonometry Examples

$\frac{x}{x^{2} - 16} - \frac{x + 5}{x^{2} - 4 x} = \frac{- 8}{x^{2} + 4 x}$

Step 1

Factor each term.

Tap for more steps...

Step 1.1

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

$\frac{x}{x^{2} - 4^{2}} - \frac{x + 5}{x^{2} - 4 x} = \frac{- 8}{x^{2} + 4 x}$

Step 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 $b = 4$ .

$\frac{x}{(x + 4) (x - 4)} - \frac{x + 5}{x^{2} - 4 x} = \frac{- 8}{x^{2} + 4 x}$

Step 1.3

Factor $x$ out of $x^{2} - 4 x$ .

Tap for more steps...

Step 1.3.1

Factor $x$ out of $x^{2}$ .

$\frac{x}{(x + 4) (x - 4)} - \frac{x + 5}{x \cdot x - 4 x} = \frac{- 8}{x^{2} + 4 x}$

Step 1.3.2

Factor $x$ out of $- 4 x$ .

$\frac{x}{(x + 4) (x - 4)} - \frac{x + 5}{x \cdot x + x \cdot - 4} = \frac{- 8}{x^{2} + 4 x}$

Step 1.3.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$\frac{x}{(x + 4) (x - 4)} - \frac{x + 5}{x (x - 4)} = \frac{- 8}{x^{2} + 4 x}$

Step 1.4

Factor $x$ out of $x^{2} + 4 x$ .

Tap for more steps...

Step 1.4.1

Factor $x$ out of $x^{2}$ .

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

Step 1.4.2

Factor $x$ out of $4 x$ .

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

Step 1.4.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

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

Step 1.5

Move the negative in front of the fraction.

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

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(x + 4) (x - 4), x (x - 4), x (x + 4)$

Step 2.2

Since $(x + 4) (x - 4), x (x - 4), x (x + 4)$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $(x + 4) (x - 4), x (x - 4), x (x + 4)$ are:

1. Find the LCM for the numeric part $1, 1, 1$ .

2. Find the LCM for the variable part $x^{1}, x^{1}$ .

3. Find the LCM for the compound variable part $x + 4, x - 4, x - 4, x + 4$ .

4. Multiply each LCM together.

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.6

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 2.7

The LCM of $x^{1}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

Step 2.8

The factor for $x + 4$ is $x + 4$ itself.

$(x + 4) = x + 4$

$(x + 4)$ occurs $1$ time.

Step 2.9

The factor for $x - 4$ is $x - 4$ itself.

$(x - 4) = x - 4$

$(x - 4)$ occurs $1$ time.

Step 2.10

The factor for $x + 4$ is $x + 4$ itself.

$(x + 4) = x + 4$

$(x + 4)$ occurs $1$ time.

Step 2.11

The LCM of $x + 4, x - 4, x - 4, x + 4$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x + 4) (x - 4)$

Step 2.12

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$x (x + 4) (x - 4)$

Step 3

Multiply each term in $\frac{x}{(x + 4) (x - 4)} - \frac{x + 5}{x (x - 4)} = - \frac{8}{x (x + 4)}$ by $x (x + 4) (x - 4)$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{x}{(x + 4) (x - 4)} - \frac{x + 5}{x (x - 4)} = - \frac{8}{x (x + 4)}$ by $x (x + 4) (x - 4)$ .

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Cancel the common factor of $(x + 4) (x - 4)$ .

Tap for more steps...

Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor.

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

Step 3.2.1.1.3

Rewrite the expression.

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

Step 3.2.1.2

Raise $x$ to the power of $1$ .

$x^{1} x - \frac{x + 5}{x (x - 4)} (x (x + 4) (x - 4)) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.3

Raise $x$ to the power of $1$ .

$x^{1} x^{1} - \frac{x + 5}{x (x - 4)} (x (x + 4) (x - 4)) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + 1} - \frac{x + 5}{x (x - 4)} (x (x + 4) (x - 4)) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.5

Add $1$ and $1$ .

$x^{2} - \frac{x + 5}{x (x - 4)} (x (x + 4) (x - 4)) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.6

Cancel the common factor of $x (x - 4)$ .

Tap for more steps...

Step 3.2.1.6.1

Move the leading negative in $- \frac{x + 5}{x (x - 4)}$ into the numerator.

$x^{2} + \frac{- (x + 5)}{x (x - 4)} (x (x + 4) (x - 4)) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.6.2

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

$x^{2} + \frac{- (x + 5)}{x (x - 4)} (x (x - 4) (x + 4)) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.6.3

Cancel the common factor.

$x^{2} + \frac{- (x + 5)}{x (x - 4)} (x (x - 4) (x + 4)) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.6.4

Rewrite the expression.

$x^{2} - (x + 5) (x + 4) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.7

Apply the distributive property.

$x^{2} + (- x - 1 \cdot 5) (x + 4) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.8

Multiply $- 1$ by $5$ .

$x^{2} + (- x - 5) (x + 4) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.9

Expand $(- x - 5) (x + 4)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.9.1

Apply the distributive property.

$x^{2} - x (x + 4) - 5 (x + 4) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.9.2

Apply the distributive property.

$x^{2} - x \cdot x - x \cdot 4 - 5 (x + 4) = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.9.3

Apply the distributive property.

$x^{2} - x \cdot x - x \cdot 4 - 5 x - 5 \cdot 4 = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.10

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.10.1

Simplify each term.

Tap for more steps...

Step 3.2.1.10.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.2.1.10.1.1.1

Move $x$ .

$x^{2} - (x \cdot x) - x \cdot 4 - 5 x - 5 \cdot 4 = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.10.1.1.2

Multiply $x$ by $x$ .

$x^{2} - x^{2} - x \cdot 4 - 5 x - 5 \cdot 4 = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.10.1.2

Multiply $4$ by $- 1$ .

$x^{2} - x^{2} - 4 x - 5 x - 5 \cdot 4 = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.10.1.3

Multiply $- 5$ by $4$ .

$x^{2} - x^{2} - 4 x - 5 x - 20 = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.1.10.2

Subtract $5 x$ from $- 4 x$ .

$x^{2} - x^{2} - 9 x - 20 = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.2

Simplify by adding terms.

Tap for more steps...

Step 3.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$0 - 9 x - 20 = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.2.2.2

Subtract $9 x$ from $0$ .

$- 9 x - 20 = - \frac{8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $x (x + 4)$ .

Tap for more steps...

Step 3.3.1.1

Move the leading negative in $- \frac{8}{x (x + 4)}$ into the numerator.

$- 9 x - 20 = \frac{- 8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.3.1.2

Cancel the common factor.

$- 9 x - 20 = \frac{- 8}{x (x + 4)} (x (x + 4) (x - 4))$

Step 3.3.1.3

Rewrite the expression.

$- 9 x - 20 = - 8 (x - 4)$

Step 3.3.2

Apply the distributive property.

$- 9 x - 20 = - 8 x - 8 \cdot - 4$

Step 3.3.3

Multiply $- 8$ by $- 4$ .

$- 9 x - 20 = - 8 x + 32$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 4.1.1

Add $8 x$ to both sides of the equation.

$- 9 x - 20 + 8 x = 32$

Step 4.1.2

Add $- 9 x$ and $8 x$ .

$- x - 20 = 32$

Step 4.2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 4.2.1

Add $20$ to both sides of the equation.

$- x = 32 + 20$

Step 4.2.2

Add $32$ and $20$ .

$- x = 52$

Step 4.3

Divide each term in $- x = 52$ by $- 1$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $- x = 52$ by $- 1$ .

$\frac{- x}{- 1} = \frac{52}{- 1}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{52}{- 1}$

Step 4.3.2.2

Divide $x$ by $1$ .

$x = \frac{52}{- 1}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Divide $52$ by $- 1$ .

$x = - 52$