Find All Complex Solutions (x^2-x-6)/(x^2)=(x-6)/(2x)+(2x+12)/x

$\frac{x^{2} - x - 6}{x^{2}} = \frac{x - 6}{2 x} + \frac{2 x + 12}{x}$

Factor each term.

Factor $x^{2} - x - 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

Factor $2$ out of $2 x + 12$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $12$ .

$\frac{(x - 3) (x + 2)}{x^{2}} = \frac{x - 6}{2 x} + \frac{2 x + 2 \cdot 6}{x}$

Factor $2$ out of $2 x + 2 \cdot 6$ .

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

Find the LCD of the terms in the equation.

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

$x^{2}, 2 x, x$

Since $x^{2}, 2 x, x$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 2, 1$ then find LCM for the variable part $x^{2}, x^{1}, x^{1}$ .

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.

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

Not prime

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

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

Not prime

The LCM of $1, 2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

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

$x^{1} = x$

$x$ occurs $1$ time.

The LCM of $x^{2}, x^{1}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Multiply $x$ by $x$ .

$x^{2}$

The LCM for $x^{2}, 2 x, x$ is the numeric part $2$ multiplied by the variable part.

$2 x^{2}$

Multiply each term in $\frac{(x - 3) (x + 2)}{x^{2}} = \frac{x - 6}{2 x} + \frac{2 (x + 6)}{x}$ by $2 x^{2}$ to eliminate the fractions.

Multiply each term in $\frac{(x - 3) (x + 2)}{x^{2}} = \frac{x - 6}{2 x} + \frac{2 (x + 6)}{x}$ by $2 x^{2}$ .

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

Simplify the left side.

Simplify terms.

Rewrite using the commutative property of multiplication.

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

Combine $2$ and $\frac{(x - 3) (x + 2)}{x^{2}}$ .

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

Cancel the common factor of $x^{2}$ .

Cancel the common factor.

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

Rewrite the expression.

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

Expand $(x - 3) (x + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$2 (x \cdot x + x \cdot 2 - 3 (x + 2)) = \frac{x - 6}{2 x} (2 x^{2}) + \frac{2 (x + 6)}{x} (2 x^{2})$

Apply the distributive property.

$2 (x \cdot x + x \cdot 2 - 3 x - 3 \cdot 2) = \frac{x - 6}{2 x} (2 x^{2}) + \frac{2 (x + 6)}{x} (2 x^{2})$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 (x^{2} + x \cdot 2 - 3 x - 3 \cdot 2) = \frac{x - 6}{2 x} (2 x^{2}) + \frac{2 (x + 6)}{x} (2 x^{2})$

Move $2$ to the left of $x$ .

$2 (x^{2} + 2 \cdot x - 3 x - 3 \cdot 2) = \frac{x - 6}{2 x} (2 x^{2}) + \frac{2 (x + 6)}{x} (2 x^{2})$

Multiply $- 3$ by $2$ .

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

Subtract $3 x$ from $2 x$ .

$2 (x^{2} - x - 6) = \frac{x - 6}{2 x} (2 x^{2}) + \frac{2 (x + 6)}{x} (2 x^{2})$

Apply the distributive property.

$2 x^{2} + 2 (- x) + 2 \cdot - 6 = \frac{x - 6}{2 x} (2 x^{2}) + \frac{2 (x + 6)}{x} (2 x^{2})$

Simplify.

Multiply $- 1$ by $2$ .

$2 x^{2} - 2 x + 2 \cdot - 6 = \frac{x - 6}{2 x} (2 x^{2}) + \frac{2 (x + 6)}{x} (2 x^{2})$

Multiply $2$ by $- 6$ .

$2 x^{2} - 2 x - 12 = \frac{x - 6}{2 x} (2 x^{2}) + \frac{2 (x + 6)}{x} (2 x^{2})$

Simplify the right side.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 x^{2} - 2 x - 12 = 2 \frac{x - 6}{2 x} x^{2} + \frac{2 (x + 6)}{x} (2 x^{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$2 x^{2} - 2 x - 12 = 2 \frac{x - 6}{2 (x)} x^{2} + \frac{2 (x + 6)}{x} (2 x^{2})$

Cancel the common factor.

$2 x^{2} - 2 x - 12 = 2 \frac{x - 6}{2 x} x^{2} + \frac{2 (x + 6)}{x} (2 x^{2})$

Rewrite the expression.

$2 x^{2} - 2 x - 12 = \frac{x - 6}{x} x^{2} + \frac{2 (x + 6)}{x} (2 x^{2})$

Cancel the common factor of $x$ .

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

$2 x^{2} - 2 x - 12 = \frac{x - 6}{x} (x \cdot x) + \frac{2 (x + 6)}{x} (2 x^{2})$

Cancel the common factor.

$2 x^{2} - 2 x - 12 = \frac{x - 6}{x} (x \cdot x) + \frac{2 (x + 6)}{x} (2 x^{2})$

Rewrite the expression.

$2 x^{2} - 2 x - 12 = (x - 6) x + \frac{2 (x + 6)}{x} (2 x^{2})$

Apply the distributive property.

$2 x^{2} - 2 x - 12 = x \cdot x - 6 x + \frac{2 (x + 6)}{x} (2 x^{2})$

Multiply $x$ by $x$ .

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + \frac{2 (x + 6)}{x} (2 x^{2})$

Rewrite using the commutative property of multiplication.

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + 2 \frac{2 (x + 6)}{x} x^{2}$

Multiply $2 \frac{2 (x + 6)}{x}$ .

Combine $2$ and $\frac{2 (x + 6)}{x}$ .

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + \frac{2 (2 (x + 6))}{x} x^{2}$

Multiply $2$ by $2$ .

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + \frac{4 (x + 6)}{x} x^{2}$

Cancel the common factor of $x$ .

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

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + \frac{4 (x + 6)}{x} (x \cdot x)$

Cancel the common factor.

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + \frac{4 (x + 6)}{x} (x \cdot x)$

Rewrite the expression.

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + 4 (x + 6) x$

Apply the distributive property.

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + (4 x + 4 \cdot 6) x$

Multiply $4$ by $6$ .

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + (4 x + 24) x$

Apply the distributive property.

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + 4 x \cdot x + 24 x$

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

Move $x$ .

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + 4 (x \cdot x) + 24 x$

Multiply $x$ by $x$ .

$2 x^{2} - 2 x - 12 = x^{2} - 6 x + 4 x^{2} + 24 x$

Simplify by adding terms.

Add $x^{2}$ and $4 x^{2}$ .

$2 x^{2} - 2 x - 12 = 5 x^{2} - 6 x + 24 x$

Add $- 6 x$ and $24 x$ .

$2 x^{2} - 2 x - 12 = 5 x^{2} + 18 x$

Solve the equation.

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

Subtract $5 x^{2}$ from both sides of the equation.

$2 x^{2} - 2 x - 12 - 5 x^{2} = 18 x$

Subtract $18 x$ from both sides of the equation.

$2 x^{2} - 2 x - 12 - 5 x^{2} - 18 x = 0$

Subtract $5 x^{2}$ from $2 x^{2}$ .

$- 3 x^{2} - 2 x - 12 - 18 x = 0$

Subtract $18 x$ from $- 2 x$ .

$- 3 x^{2} - 20 x - 12 = 0$

Factor the left side of the equation.

Factor $- 1$ out of $- 3 x^{2} - 20 x - 12$ .

Factor $- 1$ out of $- 3 x^{2}$ .

$- (3 x^{2}) - 20 x - 12 = 0$

Factor $- 1$ out of $- 20 x$ .

$- (3 x^{2}) - (20 x) - 12 = 0$

Rewrite $- 12$ as $- 1 (12)$ .

$- (3 x^{2}) - (20 x) - 1 \cdot 12 = 0$

Factor $- 1$ out of $- (3 x^{2}) - (20 x)$ .

$- (3 x^{2} + 20 x) - 1 \cdot 12 = 0$

Factor $- 1$ out of $- (3 x^{2} + 20 x) - 1 (12)$ .

$- (3 x^{2} + 20 x + 12) = 0$

Factor.

Factor by grouping.

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 $a \cdot c = 3 \cdot 12 = 36$ and whose sum is $b = 20$ .

Factor $20$ out of $20 x$ .

$- (3 x^{2} + 20 (x) + 12) = 0$

Rewrite $20$ as $2$ plus $18$

$- (3 x^{2} + (2 + 18) x + 12) = 0$

Apply the distributive property.

$- (3 x^{2} + 2 x + 18 x + 12) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((3 x^{2} + 2 x) + 18 x + 12) = 0$

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

$- (x (3 x + 2) + 6 (3 x + 2)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

$- ((3 x + 2) (x + 6)) = 0$

Remove unnecessary parentheses.

$- (3 x + 2) (x + 6) = 0$

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

$3 x + 2 = 0$

$x + 6 = 0$

Set $3 x + 2$ equal to $0$ and solve for $x$ .

Set $3 x + 2$ equal to $0$ .

$3 x + 2 = 0$

Solve $3 x + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Divide each term in $3 x = - 2$ by $3$ and simplify.

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3}$

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $- (3 x + 2) (x + 6) = 0$ true.

$x = - \frac{2}{3}, - 6$

Find All Complex Solutions (x^2-x-6)/(x^2)=(x-6)/(2x)+(2x+12)/x

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