Pre-Algebra Examples

$- 3 (x + 2 + 4) (2 x - 4) = - 2 (x - 5)$

Step 1

Simplify $- 3 (x + 2 + 4) (2 x - 4)$ .

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Step 1.1

Rewrite.

$0 + 0 - 3 (x + 2 + 4) (2 x - 4) = - 2 (x - 5)$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Add $0$ and $0$ .

$0 - 3 (x + 2 + 4) (2 x - 4) = - 2 (x - 5)$

Step 1.2.2

Add $2$ and $4$ .

$- 3 (x + 6) (2 x - 4) = - 2 (x - 5)$

Step 1.2.3

Apply the distributive property.

$(- 3 x - 3 \cdot 6) (2 x - 4) = - 2 (x - 5)$

Step 1.2.4

Multiply $- 3$ by $6$ .

$(- 3 x - 18) (2 x - 4) = - 2 (x - 5)$

Step 1.3

Expand $(- 3 x - 18) (2 x - 4)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$- 3 x (2 x - 4) - 18 (2 x - 4) = - 2 (x - 5)$

Step 1.3.2

Apply the distributive property.

$- 3 x (2 x) - 3 x \cdot - 4 - 18 (2 x - 4) = - 2 (x - 5)$

Step 1.3.3

Apply the distributive property.

$- 3 x (2 x) - 3 x \cdot - 4 - 18 (2 x) - 18 \cdot - 4 = - 2 (x - 5)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$- 3 \cdot 2 x \cdot x - 3 x \cdot - 4 - 18 (2 x) - 18 \cdot - 4 = - 2 (x - 5)$

Step 1.4.1.2

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

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Step 1.4.1.2.1

Move $x$ .

$- 3 \cdot 2 (x \cdot x) - 3 x \cdot - 4 - 18 (2 x) - 18 \cdot - 4 = - 2 (x - 5)$

Step 1.4.1.2.2

Multiply $x$ by $x$ .

$- 3 \cdot 2 x^{2} - 3 x \cdot - 4 - 18 (2 x) - 18 \cdot - 4 = - 2 (x - 5)$

Step 1.4.1.3

Multiply $- 3$ by $2$ .

$- 6 x^{2} - 3 x \cdot - 4 - 18 (2 x) - 18 \cdot - 4 = - 2 (x - 5)$

Step 1.4.1.4

Multiply $- 4$ by $- 3$ .

$- 6 x^{2} + 12 x - 18 (2 x) - 18 \cdot - 4 = - 2 (x - 5)$

Step 1.4.1.5

Multiply $2$ by $- 18$ .

$- 6 x^{2} + 12 x - 36 x - 18 \cdot - 4 = - 2 (x - 5)$

Step 1.4.1.6

Multiply $- 18$ by $- 4$ .

$- 6 x^{2} + 12 x - 36 x + 72 = - 2 (x - 5)$

Step 1.4.2

Subtract $36 x$ from $12 x$ .

$- 6 x^{2} - 24 x + 72 = - 2 (x - 5)$

Step 2

Simplify $- 2 (x - 5)$ .

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Step 2.1

Apply the distributive property.

$- 6 x^{2} - 24 x + 72 = - 2 x - 2 \cdot - 5$

Step 2.2

Multiply $- 2$ by $- 5$ .

$- 6 x^{2} - 24 x + 72 = - 2 x + 10$

Step 3

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

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Step 3.1

Add $2 x$ to both sides of the equation.

$- 6 x^{2} - 24 x + 72 + 2 x = 10$

Step 3.2

Add $- 24 x$ and $2 x$ .

$- 6 x^{2} - 22 x + 72 = 10$

Step 4

Subtract $10$ from both sides of the equation.

$- 6 x^{2} - 22 x + 72 - 10 = 0$

Step 5

Subtract $10$ from $72$ .

$- 6 x^{2} - 22 x + 62 = 0$

Step 6

Factor $- 2$ out of $- 6 x^{2} - 22 x + 62$ .

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Step 6.1

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

$- 2 (3 x^{2}) - 22 x + 62 = 0$

Step 6.2

Factor $- 2$ out of $- 22 x$ .

$- 2 (3 x^{2}) - 2 (11 x) + 62 = 0$

Step 6.3

Factor $- 2$ out of $62$ .

$- 2 (3 x^{2}) - 2 (11 x) - 2 \cdot - 31 = 0$

Step 6.4

Factor $- 2$ out of $- 2 (3 x^{2}) - 2 (11 x)$ .

$- 2 (3 x^{2} + 11 x) - 2 \cdot - 31 = 0$

Step 6.5

Factor $- 2$ out of $- 2 (3 x^{2} + 11 x) - 2 (- 31)$ .

$- 2 (3 x^{2} + 11 x - 31) = 0$

Step 7

Divide each term in $- 2 (3 x^{2} + 11 x - 31) = 0$ by $- 2$ and simplify.

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Step 7.1

Divide each term in $- 2 (3 x^{2} + 11 x - 31) = 0$ by $- 2$ .

$\frac{- 2 (3 x^{2} + 11 x - 31)}{- 2} = \frac{0}{- 2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 2 (3 x^{2} + 11 x - 31)}{- 2} = \frac{0}{- 2}$

Step 7.2.1.2

Divide $3 x^{2} + 11 x - 31$ by $1$ .

$3 x^{2} + 11 x - 31 = \frac{0}{- 2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $- 2$ .

$3 x^{2} + 11 x - 31 = 0$

Step 8

Use the quadratic formula to find the solutions.

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

Step 9

Substitute the values $a = 3$ , $b = 11$ , and $c = - 31$ into the quadratic formula and solve for $x$ .

$\frac{- 11 \pm \sqrt{11^{2} - 4 \cdot (3 \cdot - 31)}}{2 \cdot 3}$

Step 10

Simplify.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3 \cdot - 31}}{2 \cdot 3}$

Step 10.1.2

Multiply $- 4 \cdot 3 \cdot - 31$ .

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Step 10.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12 \cdot - 31}}{2 \cdot 3}$

Step 10.1.2.2

Multiply $- 12$ by $- 31$ .

$x = \frac{- 11 \pm \sqrt{121 + 372}}{2 \cdot 3}$

Step 10.1.3

Add $121$ and $372$ .

$x = \frac{- 11 \pm \sqrt{493}}{2 \cdot 3}$

Step 10.2

Multiply $2$ by $3$ .

$x = \frac{- 11 \pm \sqrt{493}}{6}$

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 $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3 \cdot - 31}}{2 \cdot 3}$

Step 11.1.2

Multiply $- 4 \cdot 3 \cdot - 31$ .

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Step 11.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12 \cdot - 31}}{2 \cdot 3}$

Step 11.1.2.2

Multiply $- 12$ by $- 31$ .

$x = \frac{- 11 \pm \sqrt{121 + 372}}{2 \cdot 3}$

Step 11.1.3

Add $121$ and $372$ .

$x = \frac{- 11 \pm \sqrt{493}}{2 \cdot 3}$

Step 11.2

Multiply $2$ by $3$ .

$x = \frac{- 11 \pm \sqrt{493}}{6}$

Step 11.3

Change the $\pm$ to $+$ .

$x = \frac{- 11 + \sqrt{493}}{6}$

Step 11.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \cdot 11 + \sqrt{493}}{6}$

Step 11.5

Factor $- 1$ out of $\sqrt{493}$ .

$x = \frac{- 1 \cdot 11 - 1 (- \sqrt{493})}{6}$

Step 11.6

Factor $- 1$ out of $- 1 (11) - 1 (- \sqrt{493})$ .

$x = \frac{- 1 (11 - \sqrt{493})}{6}$

Step 11.7

Move the negative in front of the fraction.

$x = - \frac{11 - \sqrt{493}}{6}$

Step 12

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

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3 \cdot - 31}}{2 \cdot 3}$

Step 12.1.2

Multiply $- 4 \cdot 3 \cdot - 31$ .

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Step 12.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12 \cdot - 31}}{2 \cdot 3}$

Step 12.1.2.2

Multiply $- 12$ by $- 31$ .

$x = \frac{- 11 \pm \sqrt{121 + 372}}{2 \cdot 3}$

Step 12.1.3

Add $121$ and $372$ .

$x = \frac{- 11 \pm \sqrt{493}}{2 \cdot 3}$

Step 12.2

Multiply $2$ by $3$ .

$x = \frac{- 11 \pm \sqrt{493}}{6}$

Step 12.3

Change the $\pm$ to $-$ .

$x = \frac{- 11 - \sqrt{493}}{6}$

Step 12.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \cdot 11 - \sqrt{493}}{6}$

Step 12.5

Factor $- 1$ out of $- \sqrt{493}$ .

$x = \frac{- 1 \cdot 11 - (\sqrt{493})}{6}$

Step 12.6

Factor $- 1$ out of $- 1 (11) - (\sqrt{493})$ .

$x = \frac{- 1 (11 + \sqrt{493})}{6}$

Step 12.7

Move the negative in front of the fraction.

$x = - \frac{11 + \sqrt{493}}{6}$

Step 13

The final answer is the combination of both solutions.

$x = - \frac{11 - \sqrt{493}}{6}, - \frac{11 + \sqrt{493}}{6}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{11 - \sqrt{493}}{6}, - \frac{11 + \sqrt{493}}{6}$

Decimal Form:

$x = 1.86726721 \dots, - 5.53393388 \dots$