Pre-Algebra Examples

$6 x^{2} + 4 x + \frac{11}{6} = 0$

Step 1

Multiply through by the least common denominator $6$ , then simplify.

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Step 1.1

Apply the distributive property.

$6 (6 x^{2}) + 6 (4 x) + 6 (\frac{11}{6}) = 0$

Step 1.2

Simplify.

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Step 1.2.1

Multiply $6$ by $6$ .

$36 x^{2} + 6 (4 x) + 6 (\frac{11}{6}) = 0$

Step 1.2.2

Multiply $4$ by $6$ .

$36 x^{2} + 24 x + 6 (\frac{11}{6}) = 0$

Step 1.2.3

Cancel the common factor of $6$ .

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Step 1.2.3.1

Cancel the common factor.

$36 x^{2} + 24 x + 6 (\frac{11}{6}) = 0$

Step 1.2.3.2

Rewrite the expression.

$36 x^{2} + 24 x + 11 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 36$ , $b = 24$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 24 \pm \sqrt{24^{2} - 4 \cdot (36 \cdot 11)}}{2 \cdot 36}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 36 \cdot 11}}{2 \cdot 36}$

Step 4.1.2

Multiply $- 4 \cdot 36 \cdot 11$ .

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Step 4.1.2.1

Multiply $- 4$ by $36$ .

$x = \frac{- 24 \pm \sqrt{576 - 144 \cdot 11}}{2 \cdot 36}$

Step 4.1.2.2

Multiply $- 144$ by $11$ .

$x = \frac{- 24 \pm \sqrt{576 - 1584}}{2 \cdot 36}$

Step 4.1.3

Subtract $1584$ from $576$ .

$x = \frac{- 24 \pm \sqrt{- 1008}}{2 \cdot 36}$

Step 4.1.4

Rewrite $- 1008$ as $- 1 (1008)$ .

$x = \frac{- 24 \pm \sqrt{- 1 \cdot 1008}}{2 \cdot 36}$

Step 4.1.5

Rewrite $\sqrt{- 1 (1008)}$ as $\sqrt{- 1} \cdot \sqrt{1008}$ .

$x = \frac{- 24 \pm \sqrt{- 1} \cdot \sqrt{1008}}{2 \cdot 36}$

Step 4.1.6

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

$x = \frac{- 24 \pm i \cdot \sqrt{1008}}{2 \cdot 36}$

Step 4.1.7

Rewrite $1008$ as $12^{2} \cdot 7$ .

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Step 4.1.7.1

Factor $144$ out of $1008$ .

$x = \frac{- 24 \pm i \cdot \sqrt{144 (7)}}{2 \cdot 36}$

Step 4.1.7.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 24 \pm i \cdot \sqrt{12^{2} \cdot 7}}{2 \cdot 36}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{- 24 \pm i \cdot (12 \sqrt{7})}{2 \cdot 36}$

Step 4.1.9

Move $12$ to the left of $i$ .

$x = \frac{- 24 \pm 12 i \sqrt{7}}{2 \cdot 36}$

Step 4.2

Multiply $2$ by $36$ .

$x = \frac{- 24 \pm 12 i \sqrt{7}}{72}$

Step 4.3

Simplify $\frac{- 24 \pm 12 i \sqrt{7}}{72}$ .

$x = \frac{- 2 \pm i \sqrt{7}}{6}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 36 \cdot 11}}{2 \cdot 36}$

Step 5.1.2

Multiply $- 4 \cdot 36 \cdot 11$ .

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Step 5.1.2.1

Multiply $- 4$ by $36$ .

$x = \frac{- 24 \pm \sqrt{576 - 144 \cdot 11}}{2 \cdot 36}$

Step 5.1.2.2

Multiply $- 144$ by $11$ .

$x = \frac{- 24 \pm \sqrt{576 - 1584}}{2 \cdot 36}$

Step 5.1.3

Subtract $1584$ from $576$ .

$x = \frac{- 24 \pm \sqrt{- 1008}}{2 \cdot 36}$

Step 5.1.4

Rewrite $- 1008$ as $- 1 (1008)$ .

$x = \frac{- 24 \pm \sqrt{- 1 \cdot 1008}}{2 \cdot 36}$

Step 5.1.5

Rewrite $\sqrt{- 1 (1008)}$ as $\sqrt{- 1} \cdot \sqrt{1008}$ .

$x = \frac{- 24 \pm \sqrt{- 1} \cdot \sqrt{1008}}{2 \cdot 36}$

Step 5.1.6

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

$x = \frac{- 24 \pm i \cdot \sqrt{1008}}{2 \cdot 36}$

Step 5.1.7

Rewrite $1008$ as $12^{2} \cdot 7$ .

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Step 5.1.7.1

Factor $144$ out of $1008$ .

$x = \frac{- 24 \pm i \cdot \sqrt{144 (7)}}{2 \cdot 36}$

Step 5.1.7.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 24 \pm i \cdot \sqrt{12^{2} \cdot 7}}{2 \cdot 36}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{- 24 \pm i \cdot (12 \sqrt{7})}{2 \cdot 36}$

Step 5.1.9

Move $12$ to the left of $i$ .

$x = \frac{- 24 \pm 12 i \sqrt{7}}{2 \cdot 36}$

Step 5.2

Multiply $2$ by $36$ .

$x = \frac{- 24 \pm 12 i \sqrt{7}}{72}$

Step 5.3

Simplify $\frac{- 24 \pm 12 i \sqrt{7}}{72}$ .

$x = \frac{- 2 \pm i \sqrt{7}}{6}$

Step 5.4

Change the $\pm$ to $+$ .

$x = \frac{- 2 + i \sqrt{7}}{6}$

Step 5.5

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

$x = \frac{- 1 \cdot 2 + i \sqrt{7}}{6}$

Step 5.6

Factor $- 1$ out of $i \sqrt{7}$ .

$x = \frac{- 1 \cdot 2 - (- i \sqrt{7})}{6}$

Step 5.7

Factor $- 1$ out of $- 1 (2) - (- i \sqrt{7})$ .

$x = \frac{- 1 (2 - i \sqrt{7})}{6}$

Step 5.8

Move the negative in front of the fraction.

$x = - \frac{2 - i \sqrt{7}}{6}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $24$ to the power of $2$ .

$x = \frac{- 24 \pm \sqrt{576 - 4 \cdot 36 \cdot 11}}{2 \cdot 36}$

Step 6.1.2

Multiply $- 4 \cdot 36 \cdot 11$ .

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Step 6.1.2.1

Multiply $- 4$ by $36$ .

$x = \frac{- 24 \pm \sqrt{576 - 144 \cdot 11}}{2 \cdot 36}$

Step 6.1.2.2

Multiply $- 144$ by $11$ .

$x = \frac{- 24 \pm \sqrt{576 - 1584}}{2 \cdot 36}$

Step 6.1.3

Subtract $1584$ from $576$ .

$x = \frac{- 24 \pm \sqrt{- 1008}}{2 \cdot 36}$

Step 6.1.4

Rewrite $- 1008$ as $- 1 (1008)$ .

$x = \frac{- 24 \pm \sqrt{- 1 \cdot 1008}}{2 \cdot 36}$

Step 6.1.5

Rewrite $\sqrt{- 1 (1008)}$ as $\sqrt{- 1} \cdot \sqrt{1008}$ .

$x = \frac{- 24 \pm \sqrt{- 1} \cdot \sqrt{1008}}{2 \cdot 36}$

Step 6.1.6

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

$x = \frac{- 24 \pm i \cdot \sqrt{1008}}{2 \cdot 36}$

Step 6.1.7

Rewrite $1008$ as $12^{2} \cdot 7$ .

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Step 6.1.7.1

Factor $144$ out of $1008$ .

$x = \frac{- 24 \pm i \cdot \sqrt{144 (7)}}{2 \cdot 36}$

Step 6.1.7.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 24 \pm i \cdot \sqrt{12^{2} \cdot 7}}{2 \cdot 36}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 24 \pm i \cdot (12 \sqrt{7})}{2 \cdot 36}$

Step 6.1.9

Move $12$ to the left of $i$ .

$x = \frac{- 24 \pm 12 i \sqrt{7}}{2 \cdot 36}$

Step 6.2

Multiply $2$ by $36$ .

$x = \frac{- 24 \pm 12 i \sqrt{7}}{72}$

Step 6.3

Simplify $\frac{- 24 \pm 12 i \sqrt{7}}{72}$ .

$x = \frac{- 2 \pm i \sqrt{7}}{6}$

Step 6.4

Change the $\pm$ to $-$ .

$x = \frac{- 2 - i \sqrt{7}}{6}$

Step 6.5

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

$x = \frac{- 1 \cdot 2 - i \sqrt{7}}{6}$

Step 6.6

Factor $- 1$ out of $- i \sqrt{7}$ .

$x = \frac{- 1 \cdot 2 - (i \sqrt{7})}{6}$

Step 6.7

Factor $- 1$ out of $- 1 (2) - (i \sqrt{7})$ .

$x = \frac{- 1 (2 + i \sqrt{7})}{6}$

Step 6.8

Move the negative in front of the fraction.

$x = - \frac{2 + i \sqrt{7}}{6}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{2 - i \sqrt{7}}{6}, - \frac{2 + i \sqrt{7}}{6}$