Pre-Algebra Examples

$4 x - 12 = 2 x (x + 6)$

Step 1

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

$2 x (x + 6) = 4 x - 12$

Step 2

Simplify $2 x (x + 6)$ .

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Step 2.1

Rewrite.

$0 + 0 + 2 x (x + 6) = 4 x - 12$

Step 2.2

Simplify by adding zeros.

$2 x (x + 6) = 4 x - 12$

Step 2.3

Apply the distributive property.

$2 x \cdot x + 2 x \cdot 6 = 4 x - 12$

Step 2.4

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

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Step 2.4.1

Move $x$ .

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

Step 2.4.2

Multiply $x$ by $x$ .

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

Step 2.5

Multiply $6$ by $2$ .

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

Step 3

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

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Step 3.1

Subtract $4 x$ from both sides of the equation.

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

Step 3.2

Subtract $4 x$ from $12 x$ .

$2 x^{2} + 8 x = - 12$

Step 4

Add $12$ to both sides of the equation.

$2 x^{2} + 8 x + 12 = 0$

Step 5

Factor $2$ out of $2 x^{2} + 8 x + 12$ .

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Step 5.1

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

$2 (x^{2}) + 8 x + 12 = 0$

Step 5.2

Factor $2$ out of $8 x$ .

$2 (x^{2}) + 2 (4 x) + 12 = 0$

Step 5.3

Factor $2$ out of $12$ .

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

Step 5.4

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

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

Step 5.5

Factor $2$ out of $2 (x^{2} + 4 x) + 2 \cdot 6$ .

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

Step 6

Divide each term in $2 (x^{2} + 4 x + 6) = 0$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 (x^{2} + 4 x + 6) = 0$ by $2$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x^{2} + 4 x + 6$ by $1$ .

$x^{2} + 4 x + 6 = \frac{0}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $2$ .

$x^{2} + 4 x + 6 = 0$

Step 7

Use the quadratic formula to find the solutions.

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

Step 8

Substitute the values $a = 1$ , $b = 4$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 6)}}{2 \cdot 1}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 6}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 4 \pm \sqrt{16 - 24}}{2 \cdot 1}$

Step 9.1.3

Subtract $24$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 9.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 9.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 9.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 9.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 9.1.7.1

Factor $4$ out of $8$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 9.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 9.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 9.1.9

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

$x = \frac{- 4 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 i \sqrt{2}}{2}$

Step 9.3

Simplify $\frac{- 4 \pm 2 i \sqrt{2}}{2}$ .

$x = - 2 \pm i \sqrt{2}$

Step 10

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 6}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 4 \pm \sqrt{16 - 24}}{2 \cdot 1}$

Step 10.1.3

Subtract $24$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 10.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 10.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 10.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 10.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 10.1.7.1

Factor $4$ out of $8$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 10.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 10.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 10.1.9

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

$x = \frac{- 4 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 i \sqrt{2}}{2}$

Step 10.3

Simplify $\frac{- 4 \pm 2 i \sqrt{2}}{2}$ .

$x = - 2 \pm i \sqrt{2}$

Step 10.4

Change the $\pm$ to $+$ .

$x = - 2 + i \sqrt{2}$

Step 11

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 11.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 11.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 6}}{2 \cdot 1}$

Step 11.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 4 \pm \sqrt{16 - 24}}{2 \cdot 1}$

Step 11.1.3

Subtract $24$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 11.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 11.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 11.1.6

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

$x = \frac{- 4 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 11.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 11.1.7.1

Factor $4$ out of $8$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 11.1.7.2

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

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 11.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 11.1.9

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

$x = \frac{- 4 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 11.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 i \sqrt{2}}{2}$

Step 11.3

Simplify $\frac{- 4 \pm 2 i \sqrt{2}}{2}$ .

$x = - 2 \pm i \sqrt{2}$

Step 11.4

Change the $\pm$ to $-$ .

$x = - 2 - i \sqrt{2}$

Step 12

The final answer is the combination of both solutions.

$x = - 2 + i \sqrt{2}, - 2 - i \sqrt{2}$