Pre-Algebra Examples

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

Step 1

Simplify $\frac{1}{2} \cdot (x + 2)$ .

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Step 1.1

Apply the distributive property.

$3 x^{2} + 6 x = \frac{1}{2} x + \frac{1}{2} \cdot 2$

Step 1.2

Combine $\frac{1}{2}$ and $x$ .

$3 x^{2} + 6 x = \frac{x}{2} + \frac{1}{2} \cdot 2$

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Cancel the common factor.

$3 x^{2} + 6 x = \frac{x}{2} + \frac{1}{2} \cdot 2$

Step 1.3.2

Rewrite the expression.

$3 x^{2} + 6 x = \frac{x}{2} + 1$

Step 2

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

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Step 2.1

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

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

Step 2.2

To write $6 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 x^{2} + 6 x \cdot \frac{2}{2} - \frac{x}{2} = 1$

Step 2.3

Combine $6 x$ and $\frac{2}{2}$ .

$3 x^{2} + \frac{6 x \cdot 2}{2} - \frac{x}{2} = 1$

Step 2.4

Combine the numerators over the common denominator.

$3 x^{2} + \frac{6 x \cdot 2 - x}{2} = 1$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

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Step 2.5.1.1.1

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

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

Step 2.5.1.1.2

Factor $x$ out of $- x$ .

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

Step 2.5.1.1.3

Factor $x$ out of $x (6 \cdot 2) + x \cdot - 1$ .

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

Step 2.5.1.2

Multiply $6$ by $2$ .

$3 x^{2} + \frac{x (12 - 1)}{2} = 1$

Step 2.5.1.3

Subtract $1$ from $12$ .

$3 x^{2} + \frac{x \cdot 11}{2} = 1$

Step 2.5.2

Move $11$ to the left of $x$ .

$3 x^{2} + \frac{11 x}{2} = 1$

Step 3

Subtract $1$ from both sides of the equation.

$3 x^{2} + \frac{11 x}{2} - 1 = 0$

Step 4

Multiply through by the least common denominator $2$ , then simplify.

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Step 4.1

Apply the distributive property.

$2 (3 x^{2}) + 2 (\frac{11 x}{2}) + 2 \cdot - 1 = 0$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $3$ by $2$ .

$6 x^{2} + 2 (\frac{11 x}{2}) + 2 \cdot - 1 = 0$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Cancel the common factor.

$6 x^{2} + 2 (\frac{11 x}{2}) + 2 \cdot - 1 = 0$

Step 4.2.2.2

Rewrite the expression.

$6 x^{2} + 11 x + 2 \cdot - 1 = 0$

Step 4.2.3

Multiply $2$ by $- 1$ .

$6 x^{2} + 11 x - 2 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 6$ , $b = 11$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 11 \pm \sqrt{11^{2} - 4 \cdot (6 \cdot - 2)}}{2 \cdot 6}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 6 \cdot - 2}}{2 \cdot 6}$

Step 7.1.2

Multiply $- 4 \cdot 6 \cdot - 2$ .

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Step 7.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 11 \pm \sqrt{121 - 24 \cdot - 2}}{2 \cdot 6}$

Step 7.1.2.2

Multiply $- 24$ by $- 2$ .

$x = \frac{- 11 \pm \sqrt{121 + 48}}{2 \cdot 6}$

Step 7.1.3

Add $121$ and $48$ .

$x = \frac{- 11 \pm \sqrt{169}}{2 \cdot 6}$

Step 7.1.4

Rewrite $169$ as $13^{2}$ .

$x = \frac{- 11 \pm \sqrt{13^{2}}}{2 \cdot 6}$

Step 7.1.5

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

$x = \frac{- 11 \pm 13}{2 \cdot 6}$

Step 7.2

Multiply $2$ by $6$ .

$x = \frac{- 11 \pm 13}{12}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 6 \cdot - 2}}{2 \cdot 6}$

Step 8.1.2

Multiply $- 4 \cdot 6 \cdot - 2$ .

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Step 8.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 11 \pm \sqrt{121 - 24 \cdot - 2}}{2 \cdot 6}$

Step 8.1.2.2

Multiply $- 24$ by $- 2$ .

$x = \frac{- 11 \pm \sqrt{121 + 48}}{2 \cdot 6}$

Step 8.1.3

Add $121$ and $48$ .

$x = \frac{- 11 \pm \sqrt{169}}{2 \cdot 6}$

Step 8.1.4

Rewrite $169$ as $13^{2}$ .

$x = \frac{- 11 \pm \sqrt{13^{2}}}{2 \cdot 6}$

Step 8.1.5

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

$x = \frac{- 11 \pm 13}{2 \cdot 6}$

Step 8.2

Multiply $2$ by $6$ .

$x = \frac{- 11 \pm 13}{12}$

Step 8.3

Change the $\pm$ to $+$ .

$x = \frac{- 11 + 13}{12}$

Step 8.4

Add $- 11$ and $13$ .

$x = \frac{2}{12}$

Step 8.5

Cancel the common factor of $2$ and $12$ .

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Step 8.5.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{12}$

Step 8.5.2

Cancel the common factors.

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Step 8.5.2.1

Factor $2$ out of $12$ .

$x = \frac{2 \cdot 1}{2 \cdot 6}$

Step 8.5.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot 6}$

Step 8.5.2.3

Rewrite the expression.

$x = \frac{1}{6}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 6 \cdot - 2}}{2 \cdot 6}$

Step 9.1.2

Multiply $- 4 \cdot 6 \cdot - 2$ .

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Step 9.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 11 \pm \sqrt{121 - 24 \cdot - 2}}{2 \cdot 6}$

Step 9.1.2.2

Multiply $- 24$ by $- 2$ .

$x = \frac{- 11 \pm \sqrt{121 + 48}}{2 \cdot 6}$

Step 9.1.3

Add $121$ and $48$ .

$x = \frac{- 11 \pm \sqrt{169}}{2 \cdot 6}$

Step 9.1.4

Rewrite $169$ as $13^{2}$ .

$x = \frac{- 11 \pm \sqrt{13^{2}}}{2 \cdot 6}$

Step 9.1.5

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

$x = \frac{- 11 \pm 13}{2 \cdot 6}$

Step 9.2

Multiply $2$ by $6$ .

$x = \frac{- 11 \pm 13}{12}$

Step 9.3

Change the $\pm$ to $-$ .

$x = \frac{- 11 - 13}{12}$

Step 9.4

Subtract $13$ from $- 11$ .

$x = \frac{- 24}{12}$

Step 9.5

Divide $- 24$ by $12$ .

$x = - 2$

Step 10

The final answer is the combination of both solutions.

$x = \frac{1}{6}, - 2$