Trigonometry Examples

$\frac{12}{x} + \frac{18}{x + 1} = 12$

Step 1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 1.1

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

$x, x + 1, 1$

Step 1.2

Since $x, x + 1, 1$ 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 $x, x + 1, 1$ are:

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

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

3. Find the LCM for the compound variable part $x + 1$ .

4. Multiply each LCM together.

Step 1.3

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 1.4

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

Not prime

Step 1.5

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 1.6

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

$x^{1} = x$

$x$ occurs $1$ time.

Step 1.7

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

$x$

Step 1.8

The factor for $x + 1$ is $x + 1$ itself.

$(x + 1) = x + 1$

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

Step 1.9

The LCM of $x + 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$x + 1$

Step 1.10

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

$x (x + 1)$

Step 2

Multiply each term in $\frac{12}{x} + \frac{18}{x + 1} = 12$ by $x (x + 1)$ to eliminate the fractions.

Tap for more steps...

Step 2.1

Multiply each term in $\frac{12}{x} + \frac{18}{x + 1} = 12$ by $x (x + 1)$ .

$\frac{12}{x} (x (x + 1)) + \frac{18}{x + 1} (x (x + 1)) = 12 (x (x + 1))$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.1.1.1

Cancel the common factor.

$\frac{12}{x} (x (x + 1)) + \frac{18}{x + 1} (x (x + 1)) = 12 (x (x + 1))$

Step 2.2.1.1.2

Rewrite the expression.

$12 (x + 1) + \frac{18}{x + 1} (x (x + 1)) = 12 (x (x + 1))$

Step 2.2.1.2

Apply the distributive property.

$12 x + 12 \cdot 1 + \frac{18}{x + 1} (x (x + 1)) = 12 (x (x + 1))$

Step 2.2.1.3

Multiply $12$ by $1$ .

$12 x + 12 + \frac{18}{x + 1} (x (x + 1)) = 12 (x (x + 1))$

Step 2.2.1.4

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 2.2.1.4.1

Factor $x + 1$ out of $x (x + 1)$ .

$12 x + 12 + \frac{18}{x + 1} ((x + 1) x) = 12 (x (x + 1))$

Step 2.2.1.4.2

Cancel the common factor.

$12 x + 12 + \frac{18}{x + 1} ((x + 1) x) = 12 (x (x + 1))$

Step 2.2.1.4.3

Rewrite the expression.

$12 x + 12 + 18 x = 12 (x (x + 1))$

Step 2.2.2

Add $12 x$ and $18 x$ .

$30 x + 12 = 12 (x (x + 1))$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Apply the distributive property.

$30 x + 12 = 12 (x \cdot x + x \cdot 1)$

Step 2.3.2

Simplify the expression.

Tap for more steps...

Step 2.3.2.1

Multiply $x$ by $x$ .

$30 x + 12 = 12 (x^{2} + x \cdot 1)$

Step 2.3.2.2

Multiply $x$ by $1$ .

$30 x + 12 = 12 (x^{2} + x)$

Step 2.3.3

Apply the distributive property.

$30 x + 12 = 12 x^{2} + 12 x$

Step 3

Solve the equation.

Tap for more steps...

Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$12 x^{2} + 12 x = 30 x + 12$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Subtract $30 x$ from both sides of the equation.

$12 x^{2} + 12 x - 30 x = 12$

Step 3.2.2

Subtract $30 x$ from $12 x$ .

$12 x^{2} - 18 x = 12$

Step 3.3

Subtract $12$ from both sides of the equation.

$12 x^{2} - 18 x - 12 = 0$

Step 3.4

Factor the left side of the equation.

Tap for more steps...

Step 3.4.1

Factor $6$ out of $12 x^{2} - 18 x - 12$ .

Tap for more steps...

Step 3.4.1.1

Factor $6$ out of $12 x^{2}$ .

$6 (2 x^{2}) - 18 x - 12 = 0$

Step 3.4.1.2

Factor $6$ out of $- 18 x$ .

$6 (2 x^{2}) + 6 (- 3 x) - 12 = 0$

Step 3.4.1.3

Factor $6$ out of $- 12$ .

$6 (2 x^{2}) + 6 (- 3 x) + 6 (- 2) = 0$

Step 3.4.1.4

Factor $6$ out of $6 (2 x^{2}) + 6 (- 3 x)$ .

$6 (2 x^{2} - 3 x) + 6 (- 2) = 0$

Step 3.4.1.5

Factor $6$ out of $6 (2 x^{2} - 3 x) + 6 (- 2)$ .

$6 (2 x^{2} - 3 x - 2) = 0$

Step 3.4.2

Factor.

Tap for more steps...

Step 3.4.2.1

Factor by grouping.

Tap for more steps...

Step 3.4.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

Tap for more steps...

Step 3.4.2.1.1.1

Factor $- 3$ out of $- 3 x$ .

$6 (2 x^{2} - 3 x - 2) = 0$

Step 3.4.2.1.1.2

Rewrite $- 3$ as $1$ plus $- 4$

$6 (2 x^{2} + (1 - 4) x - 2) = 0$

Step 3.4.2.1.1.3

Apply the distributive property.

$6 (2 x^{2} + 1 x - 4 x - 2) = 0$

Step 3.4.2.1.1.4

Multiply $x$ by $1$ .

$6 (2 x^{2} + x - 4 x - 2) = 0$

Step 3.4.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.4.2.1.2.1

Group the first two terms and the last two terms.

$6 ((2 x^{2} + x) - 4 x - 2) = 0$

Step 3.4.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$6 (x (2 x + 1) - 2 (2 x + 1)) = 0$

Step 3.4.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$6 ((2 x + 1) (x - 2)) = 0$

Step 3.4.2.2

Remove unnecessary parentheses.

$6 (2 x + 1) (x - 2) = 0$

Step 3.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 1 = 0$

$x - 2 = 0$

Step 3.6

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.6.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 3.6.2

Solve $2 x + 1 = 0$ for $x$ .

Tap for more steps...

Step 3.6.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 3.6.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

Tap for more steps...

Step 3.6.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 3.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 3.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 3.6.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 3.7

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.7.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.8

The final solution is all the values that make $6 (2 x + 1) (x - 2) = 0$ true.

$x = - \frac{1}{2}, 2$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, 2$

Decimal Form:

$x = - 0.5, 2$