Algebra Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{x - 6}{2 x}$ from both sides of the equation.

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

Step 1.2

Subtract $\frac{2 x + 12}{x}$ from both sides of the equation.

$\frac{x^{2} - x - 6}{x^{2}} - \frac{x - 6}{2 x} - \frac{2 x + 12}{x} = 0$

Step 2

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

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{x^{2} - x - 6}{x^{2}}$ by $\frac{2}{2}$ .

$\frac{x^{2} - x - 6}{x^{2}} \cdot \frac{2}{2} - \frac{x - 6}{2 x} - \frac{2 x + 12}{x} = 0$

Step 2.1.2

Multiply $\frac{x^{2} - x - 6}{x^{2}}$ by $\frac{2}{2}$ .

$\frac{(x^{2} - x - 6) \cdot 2}{x^{2} \cdot 2} - \frac{x - 6}{2 x} - \frac{2 x + 12}{x} = 0$

Step 2.1.3

Multiply $\frac{x - 6}{2 x}$ by $\frac{x}{x}$ .

$\frac{(x^{2} - x - 6) \cdot 2}{x^{2} \cdot 2} - (\frac{x - 6}{2 x} \cdot \frac{x}{x}) - \frac{2 x + 12}{x} = 0$

Step 2.1.4

Multiply $\frac{x - 6}{2 x}$ by $\frac{x}{x}$ .

$\frac{(x^{2} - x - 6) \cdot 2}{x^{2} \cdot 2} - \frac{(x - 6) x}{2 x \cdot x} - \frac{2 x + 12}{x} = 0$

Step 2.1.5

Multiply $\frac{2 x + 12}{x}$ by $\frac{2 x}{2 x}$ .

$\frac{(x^{2} - x - 6) \cdot 2}{x^{2} \cdot 2} - \frac{(x - 6) x}{2 x \cdot x} - (\frac{2 x + 12}{x} \cdot \frac{2 x}{2 x}) = 0$

Step 2.1.6

Multiply $\frac{2 x + 12}{x}$ by $\frac{2 x}{2 x}$ .

$\frac{(x^{2} - x - 6) \cdot 2}{x^{2} \cdot 2} - \frac{(x - 6) x}{2 x \cdot x} - \frac{(2 x + 12) (2 x)}{x (2 x)} = 0$

Step 2.1.7

Reorder the factors of $x^{2} \cdot 2$ .

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 x \cdot x} - \frac{(2 x + 12) (2 x)}{x (2 x)} = 0$

Step 2.1.8

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

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 (x^{1} x)} - \frac{(2 x + 12) (2 x)}{x (2 x)} = 0$

Step 2.1.9

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

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 (x^{1} x^{1})} - \frac{(2 x + 12) (2 x)}{x (2 x)} = 0$

Step 2.1.10

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

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 x^{1 + 1}} - \frac{(2 x + 12) (2 x)}{x (2 x)} = 0$

Step 2.1.11

Add $1$ and $1$ .

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 x^{2}} - \frac{(2 x + 12) (2 x)}{x (2 x)} = 0$

Step 2.1.12

Reorder the factors of $x (2 x)$ .

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 x^{2}} - \frac{(2 x + 12) (2 x)}{2 x \cdot x} = 0$

Step 2.1.13

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

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 x^{2}} - \frac{(2 x + 12) (2 x)}{2 (x^{1} x)} = 0$

Step 2.1.14

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

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 x^{2}} - \frac{(2 x + 12) (2 x)}{2 (x^{1} x^{1})} = 0$

Step 2.1.15

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

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 x^{2}} - \frac{(2 x + 12) (2 x)}{2 x^{1 + 1}} = 0$

Step 2.1.16

Add $1$ and $1$ .

$\frac{(x^{2} - x - 6) \cdot 2}{2 x^{2}} - \frac{(x - 6) x}{2 x^{2}} - \frac{(2 x + 12) (2 x)}{2 x^{2}} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{(x^{2} - x - 6) \cdot 2 - (2 x + 12) (2 x)}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{x^{2} \cdot 2 - x \cdot 2 - 6 \cdot 2 - (2 x + 12) (2 x)}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move $2$ to the left of $x^{2}$ .

$\frac{2 \cdot x^{2} - x \cdot 2 - 6 \cdot 2 - (2 x + 12) (2 x)}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.2.2

Multiply $2$ by $- 1$ .

$\frac{2 \cdot x^{2} - 2 x - 6 \cdot 2 - (2 x + 12) (2 x)}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.2.3

Multiply $- 6$ by $2$ .

$\frac{2 \cdot x^{2} - 2 x - 12 - (2 x + 12) (2 x)}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.3

Apply the distributive property.

$\frac{2 x^{2} - 2 x - 12 + (- (2 x) - 1 \cdot 12) (2 x)}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.4

Multiply $2$ by $- 1$ .

$\frac{2 x^{2} - 2 x - 12 + (- 2 x - 1 \cdot 12) (2 x)}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.5

Multiply $- 1$ by $12$ .

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

Step 2.3.6

Apply the distributive property.

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

Step 2.3.7

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} - 2 x - 12 - 2 \cdot 2 x \cdot x - 12 (2 x)}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.8

Multiply $2$ by $- 12$ .

$\frac{2 x^{2} - 2 x - 12 - 2 \cdot 2 x \cdot x - 24 x}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.9

Simplify each term.

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Step 2.3.9.1

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

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Step 2.3.9.1.1

Move $x$ .

$\frac{2 x^{2} - 2 x - 12 - 2 \cdot 2 (x \cdot x) - 24 x}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.9.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 2 x - 12 - 2 \cdot 2 x^{2} - 24 x}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.3.9.2

Multiply $- 2$ by $2$ .

$\frac{2 x^{2} - 2 x - 12 - 4 x^{2} - 24 x}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.4

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

$\frac{- 2 x^{2} - 2 x - 12 - 24 x}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.5

Subtract $24 x$ from $- 2 x$ .

$\frac{- 2 x^{2} - 26 x - 12}{2 x^{2}} + \frac{- (x - 6) x}{2 x^{2}} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{- 2 x^{2} - 26 x - 12 - (x - 6) x}{2 x^{2}} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Apply the distributive property.

$\frac{- 2 x^{2} - 26 x - 12 + (- x - - 6) x}{2 x^{2}} = 0$

Step 2.7.1.2

Multiply $- 1$ by $- 6$ .

$\frac{- 2 x^{2} - 26 x - 12 + (- x + 6) x}{2 x^{2}} = 0$

Step 2.7.1.3

Apply the distributive property.

$\frac{- 2 x^{2} - 26 x - 12 - x \cdot x + 6 x}{2 x^{2}} = 0$

Step 2.7.1.4

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

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Step 2.7.1.4.1

Move $x$ .

$\frac{- 2 x^{2} - 26 x - 12 - (x \cdot x) + 6 x}{2 x^{2}} = 0$

Step 2.7.1.4.2

Multiply $x$ by $x$ .

$\frac{- 2 x^{2} - 26 x - 12 - x^{2} + 6 x}{2 x^{2}} = 0$

Step 2.7.2

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

$\frac{- 3 x^{2} - 26 x - 12 + 6 x}{2 x^{2}} = 0$

Step 2.7.3

Add $- 26 x$ and $6 x$ .

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

Step 2.8

Factor by grouping.

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Step 2.8.1

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

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Step 2.8.1.1

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

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

Step 2.8.1.2

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

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

Step 2.8.1.3

Apply the distributive property.

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

Step 2.8.2

Factor out the greatest common factor from each group.

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Step 2.8.2.1

Group the first two terms and the last two terms.

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

Step 2.8.2.2

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

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

Step 2.8.3

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

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

Step 2.9

Factor $- 1$ out of $- 3 x$ .

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

Step 2.10

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{(- (3 x) - 1 (2)) (x + 6)}{2 x^{2}} = 0$

Step 2.11

Factor $- 1$ out of $- (3 x) - 1 (2)$ .

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

Step 2.12

Rewrite $- (3 x + 2)$ as $- 1 (3 x + 2)$ .

$\frac{- 1 (3 x + 2) (x + 6)}{2 x^{2}} = 0$

Step 2.13

Move the negative in front of the fraction.

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

Step 3

Set the denominator in $\frac{(3 x + 2) (x + 6)}{2 x^{2}}$ equal to $0$ to find where the expression is undefined.

$2 x^{2} = 0$

Step 4

Solve for $x$ .

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Step 4.1

Divide each term in $2 x^{2} = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 x^{2} = 0$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$x^{2} = 0$

Step 4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 4.3

Simplify $\pm \sqrt{0}$ .

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Step 4.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 4.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5