Finite Math Examples

$\frac{8 + x}{15 + x} + \frac{3}{9 + 3 x} = \frac{4}{15}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Split the fraction $\frac{8 + x}{15 + x}$ into two fractions.

$\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{3}{9 + 3 x} = \frac{4}{15}$

Step 1.1.1.2

Cancel the common factor of $3$ and $9 + 3 x$ .

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Step 1.1.1.2.1

Factor $3$ out of $3$ .

$\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{3 \cdot 1}{9 + 3 x} = \frac{4}{15}$

Step 1.1.1.2.2

Cancel the common factors.

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Step 1.1.1.2.2.1

Factor $3$ out of $9$ .

$\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{3 \cdot 1}{3 (3) + 3 x} = \frac{4}{15}$

Step 1.1.1.2.2.2

Factor $3$ out of $3 x$ .

$\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{3 \cdot 1}{3 (3) + 3 (x)} = \frac{4}{15}$

Step 1.1.1.2.2.3

Factor $3$ out of $3 (3) + 3 (x)$ .

$\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{3 \cdot 1}{3 (3 + x)} = \frac{4}{15}$

Step 1.1.1.2.2.4

Cancel the common factor.

$\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{3 \cdot 1}{3 (3 + x)} = \frac{4}{15}$

Step 1.1.1.2.2.5

Rewrite the expression.

$\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{1}{3 + x} = \frac{4}{15}$

Step 1.2

Subtract $\frac{4}{15}$ from both sides of the equation.

$\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{1}{3 + x} - \frac{4}{15} = 0$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$15 + x, 15 + x, 3 + x, 15, 1$

Step 2.2

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 2.3

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

Not prime

Step 2.4

$15$ has factors of $3$ and $5$ .

$3 \cdot 5$

Step 2.5

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

Not prime

Step 2.6

The LCM of $1, 1, 1, 15, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3 \cdot 5$

Step 2.7

Multiply $3$ by $5$ .

$15$

Step 2.8

The factor for $15 + x$ is $15 + x$ itself.

$(15 + x) = 15 + x$

$(15 + x)$ occurs $1$ time.

Step 2.9

The factor for $3 + x$ is $3 + x$ itself.

$(3 + x) = 3 + x$

$(3 + x)$ occurs $1$ time.

Step 2.10

The LCM of $15 + x, 15 + x, 3 + x$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(15 + x) (3 + x)$

Step 2.11

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

$15 (15 + x) (3 + x)$

Step 3

Multiply each term in $\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{1}{3 + x} - \frac{4}{15} = 0$ by $15 (15 + x) (3 + x)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{8}{15 + x} + \frac{x}{15 + x} + \frac{1}{3 + x} - \frac{4}{15} = 0$ by $15 (15 + x) (3 + x)$ .

$\frac{8}{15 + x} (15 (15 + x) (3 + x)) + \frac{x}{15 + x} (15 (15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Rewrite using the commutative property of multiplication.

$15 \frac{8}{15 + x} ((15 + x) (3 + x)) + \frac{x}{15 + x} (15 (15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.2

Multiply $15 \frac{8}{15 + x}$ .

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Step 3.2.1.2.1

Combine $15$ and $\frac{8}{15 + x}$ .

$\frac{15 \cdot 8}{15 + x} ((15 + x) (3 + x)) + \frac{x}{15 + x} (15 (15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.2.2

Multiply $15$ by $8$ .

$\frac{120}{15 + x} ((15 + x) (3 + x)) + \frac{x}{15 + x} (15 (15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.3

Cancel the common factor of $15 + x$ .

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Step 3.2.1.3.1

Cancel the common factor.

$\frac{120}{15 + x} ((15 + x) (3 + x)) + \frac{x}{15 + x} (15 (15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.3.2

Rewrite the expression.

$120 (3 + x) + \frac{x}{15 + x} (15 (15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.4

Apply the distributive property.

$120 \cdot 3 + 120 x + \frac{x}{15 + x} (15 (15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.5

Multiply $120$ by $3$ .

$360 + 120 x + \frac{x}{15 + x} (15 (15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.6

Rewrite using the commutative property of multiplication.

$360 + 120 x + 15 \frac{x}{15 + x} ((15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.7

Combine $15$ and $\frac{x}{15 + x}$ .

$360 + 120 x + \frac{15 x}{15 + x} ((15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.8

Cancel the common factor of $15 + x$ .

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Step 3.2.1.8.1

Cancel the common factor.

$360 + 120 x + \frac{15 x}{15 + x} ((15 + x) (3 + x)) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.8.2

Rewrite the expression.

$360 + 120 x + 15 x (3 + x) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.9

Apply the distributive property.

$360 + 120 x + 15 x \cdot 3 + 15 x \cdot x + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.10

Multiply $3$ by $15$ .

$360 + 120 x + 45 x + 15 x \cdot x + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.11

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

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Step 3.2.1.11.1

Move $x$ .

$360 + 120 x + 45 x + 15 (x \cdot x) + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.11.2

Multiply $x$ by $x$ .

$360 + 120 x + 45 x + 15 x^{2} + \frac{1}{3 + x} (15 (15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.12

Rewrite using the commutative property of multiplication.

$360 + 120 x + 45 x + 15 x^{2} + 15 \frac{1}{3 + x} ((15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.13

Combine $15$ and $\frac{1}{3 + x}$ .

$360 + 120 x + 45 x + 15 x^{2} + \frac{15}{3 + x} ((15 + x) (3 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.14

Cancel the common factor of $3 + x$ .

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Step 3.2.1.14.1

Factor $3 + x$ out of $(15 + x) (3 + x)$ .

$360 + 120 x + 45 x + 15 x^{2} + \frac{15}{3 + x} ((3 + x) (15 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.14.2

Cancel the common factor.

$360 + 120 x + 45 x + 15 x^{2} + \frac{15}{3 + x} ((3 + x) (15 + x)) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.14.3

Rewrite the expression.

$360 + 120 x + 45 x + 15 x^{2} + 15 (15 + x) - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.15

Apply the distributive property.

$360 + 120 x + 45 x + 15 x^{2} + 15 \cdot 15 + 15 x - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.16

Multiply $15$ by $15$ .

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - \frac{4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.17

Cancel the common factor of $15$ .

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Step 3.2.1.17.1

Move the leading negative in $- \frac{4}{15}$ into the numerator.

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x + \frac{- 4}{15} (15 (15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.17.2

Factor $15$ out of $15 (15 + x) (3 + x)$ .

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x + \frac{- 4}{15} (15 ((15 + x) (3 + x))) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.17.3

Cancel the common factor.

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x + \frac{- 4}{15} (15 ((15 + x) (3 + x))) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.17.4

Rewrite the expression.

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 ((15 + x) (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.18

Expand $(15 + x) (3 + x)$ using the FOIL Method.

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Step 3.2.1.18.1

Apply the distributive property.

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 (15 (3 + x) + x (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.18.2

Apply the distributive property.

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 (15 \cdot 3 + 15 x + x (3 + x)) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.18.3

Apply the distributive property.

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 (15 \cdot 3 + 15 x + x \cdot 3 + x \cdot x) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.19

Simplify and combine like terms.

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Step 3.2.1.19.1

Simplify each term.

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Step 3.2.1.19.1.1

Multiply $15$ by $3$ .

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 (45 + 15 x + x \cdot 3 + x \cdot x) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.19.1.2

Move $3$ to the left of $x$ .

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 (45 + 15 x + 3 \cdot x + x \cdot x) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.19.1.3

Multiply $x$ by $x$ .

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 (45 + 15 x + 3 x + x^{2}) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.19.2

Add $15 x$ and $3 x$ .

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 (45 + 18 x + x^{2}) = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.20

Apply the distributive property.

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 4 \cdot 45 - 4 (18 x) - 4 x^{2} = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.21

Simplify.

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Step 3.2.1.21.1

Multiply $- 4$ by $45$ .

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 180 - 4 (18 x) - 4 x^{2} = 0 (15 (15 + x) (3 + x))$

Step 3.2.1.21.2

Multiply $18$ by $- 4$ .

$360 + 120 x + 45 x + 15 x^{2} + 225 + 15 x - 180 - 72 x - 4 x^{2} = 0 (15 (15 + x) (3 + x))$

Step 3.2.2

Simplify by adding terms.

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Step 3.2.2.1

Add $360$ and $225$ .

$120 x + 45 x + 15 x^{2} + 585 + 15 x - 180 - 72 x - 4 x^{2} = 0 (15 (15 + x) (3 + x))$

Step 3.2.2.2

Add $120 x$ and $45 x$ .

$165 x + 15 x^{2} + 585 + 15 x - 180 - 72 x - 4 x^{2} = 0 (15 (15 + x) (3 + x))$

Step 3.2.2.3

Add $165 x$ and $15 x$ .

$180 x + 15 x^{2} + 585 - 180 - 72 x - 4 x^{2} = 0 (15 (15 + x) (3 + x))$

Step 3.2.2.4

Subtract $72 x$ from $180 x$ .

$108 x + 15 x^{2} + 585 - 180 - 4 x^{2} = 0 (15 (15 + x) (3 + x))$

Step 3.2.2.5

Subtract $4 x^{2}$ from $15 x^{2}$ .

$108 x + 11 x^{2} + 585 - 180 = 0 (15 (15 + x) (3 + x))$

Step 3.2.2.6

Subtract $180$ from $585$ .

$108 x + 11 x^{2} + 405 = 0 (15 (15 + x) (3 + x))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify by multiplying through.

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Step 3.3.1.1

Apply the distributive property.

$108 x + 11 x^{2} + 405 = 0 ((15 \cdot 15 + 15 x) (3 + x))$

Step 3.3.1.2

Multiply $15$ by $15$ .

$108 x + 11 x^{2} + 405 = 0 ((225 + 15 x) (3 + x))$

Step 3.3.2

Expand $(225 + 15 x) (3 + x)$ using the FOIL Method.

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Step 3.3.2.1

Apply the distributive property.

$108 x + 11 x^{2} + 405 = 0 (225 (3 + x) + 15 x (3 + x))$

Step 3.3.2.2

Apply the distributive property.

$108 x + 11 x^{2} + 405 = 0 (225 \cdot 3 + 225 x + 15 x (3 + x))$

Step 3.3.2.3

Apply the distributive property.

$108 x + 11 x^{2} + 405 = 0 (225 \cdot 3 + 225 x + 15 x \cdot 3 + 15 x \cdot x)$

Step 3.3.3

Simplify and combine like terms.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply $225$ by $3$ .

$108 x + 11 x^{2} + 405 = 0 (675 + 225 x + 15 x \cdot 3 + 15 x \cdot x)$

Step 3.3.3.1.2

Multiply $3$ by $15$ .

$108 x + 11 x^{2} + 405 = 0 (675 + 225 x + 45 x + 15 x \cdot x)$

Step 3.3.3.1.3

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

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Step 3.3.3.1.3.1

Move $x$ .

$108 x + 11 x^{2} + 405 = 0 (675 + 225 x + 45 x + 15 (x \cdot x))$

Step 3.3.3.1.3.2

Multiply $x$ by $x$ .

$108 x + 11 x^{2} + 405 = 0 (675 + 225 x + 45 x + 15 x^{2})$

Step 3.3.3.2

Add $225 x$ and $45 x$ .

$108 x + 11 x^{2} + 405 = 0 (675 + 270 x + 15 x^{2})$

Step 3.3.4

Multiply $0$ by $675 + 270 x + 15 x^{2}$ .

$108 x + 11 x^{2} + 405 = 0$

Step 4

Solve the equation.

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Step 4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2

Substitute the values $a = 11$ , $b = 108$ , and $c = 405$ into the quadratic formula and solve for $x$ .

$\frac{- 108 \pm \sqrt{108^{2} - 4 \cdot (11 \cdot 405)}}{2 \cdot 11}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Raise $108$ to the power of $2$ .

$x = \frac{- 108 \pm \sqrt{11664 - 4 \cdot 11 \cdot 405}}{2 \cdot 11}$

Step 4.3.1.2

Multiply $- 4 \cdot 11 \cdot 405$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $11$ .

$x = \frac{- 108 \pm \sqrt{11664 - 44 \cdot 405}}{2 \cdot 11}$

Step 4.3.1.2.2

Multiply $- 44$ by $405$ .

$x = \frac{- 108 \pm \sqrt{11664 - 17820}}{2 \cdot 11}$

Step 4.3.1.3

Subtract $17820$ from $11664$ .

$x = \frac{- 108 \pm \sqrt{- 6156}}{2 \cdot 11}$

Step 4.3.1.4

Rewrite $- 6156$ as $- 1 (6156)$ .

$x = \frac{- 108 \pm \sqrt{- 1 \cdot 6156}}{2 \cdot 11}$

Step 4.3.1.5

Rewrite $\sqrt{- 1 (6156)}$ as $\sqrt{- 1} \cdot \sqrt{6156}$ .

$x = \frac{- 108 \pm \sqrt{- 1} \cdot \sqrt{6156}}{2 \cdot 11}$

Step 4.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 108 \pm i \cdot \sqrt{6156}}{2 \cdot 11}$

Step 4.3.1.7

Rewrite $6156$ as $18^{2} \cdot 19$ .

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Step 4.3.1.7.1

Factor $324$ out of $6156$ .

$x = \frac{- 108 \pm i \cdot \sqrt{324 (19)}}{2 \cdot 11}$

Step 4.3.1.7.2

Rewrite $324$ as $18^{2}$ .

$x = \frac{- 108 \pm i \cdot \sqrt{18^{2} \cdot 19}}{2 \cdot 11}$

Step 4.3.1.8

Pull terms out from under the radical.

$x = \frac{- 108 \pm i \cdot (18 \sqrt{19})}{2 \cdot 11}$

Step 4.3.1.9

Move $18$ to the left of $i$ .

$x = \frac{- 108 \pm 18 i \sqrt{19}}{2 \cdot 11}$

Step 4.3.2

Multiply $2$ by $11$ .

$x = \frac{- 108 \pm 18 i \sqrt{19}}{22}$

Step 4.3.3

Simplify $\frac{- 108 \pm 18 i \sqrt{19}}{22}$ .

$x = \frac{- 54 \pm 9 i \sqrt{19}}{11}$

Step 4.4

The final answer is the combination of both solutions.

$x = - \frac{54 - 9 i \sqrt{19}}{11}, - \frac{54 + 9 i \sqrt{19}}{11}$

$x = \frac{- 54 \pm 9 i \sqrt{19}}{11}$