Pre-Algebra Examples

$\frac{3}{4} x^{2} - x + \frac{19}{12} = 0$

Step 1

Combine $\frac{3}{4}$ and $x^{2}$ .

$\frac{3 x^{2}}{4} - x + \frac{19}{12} = 0$

Step 2

Multiply through by the least common denominator $12$ , then simplify.

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Step 2.1

Apply the distributive property.

$12 (\frac{3 x^{2}}{4}) + 12 (- x) + 12 (\frac{19}{12}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Factor $4$ out of $12$ .

$4 (3) (\frac{3 x^{2}}{4}) + 12 (- x) + 12 (\frac{19}{12}) = 0$

Step 2.2.1.2

Cancel the common factor.

$4 \cdot (3 (\frac{3 x^{2}}{4})) + 12 (- x) + 12 (\frac{19}{12}) = 0$

Step 2.2.1.3

Rewrite the expression.

$3 (3 x^{2}) + 12 (- x) + 12 (\frac{19}{12}) = 0$

Step 2.2.2

Multiply $3$ by $3$ .

$9 x^{2} + 12 (- x) + 12 (\frac{19}{12}) = 0$

Step 2.2.3

Multiply $- 1$ by $12$ .

$9 x^{2} - 12 x + 12 (\frac{19}{12}) = 0$

Step 2.2.4

Cancel the common factor of $12$ .

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Step 2.2.4.1

Cancel the common factor.

$9 x^{2} - 12 x + 12 (\frac{19}{12}) = 0$

Step 2.2.4.2

Rewrite the expression.

$9 x^{2} - 12 x + 19 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 9$ , $b = - 12$ , and $c = 19$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (9 \cdot 19)}}{2 \cdot 9}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 9 \cdot 19}}{2 \cdot 9}$

Step 5.1.2

Multiply $- 4 \cdot 9 \cdot 19$ .

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Step 5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{12 \pm \sqrt{144 - 36 \cdot 19}}{2 \cdot 9}$

Step 5.1.2.2

Multiply $- 36$ by $19$ .

$x = \frac{12 \pm \sqrt{144 - 684}}{2 \cdot 9}$

Step 5.1.3

Subtract $684$ from $144$ .

$x = \frac{12 \pm \sqrt{- 540}}{2 \cdot 9}$

Step 5.1.4

Rewrite $- 540$ as $- 1 (540)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 540}}{2 \cdot 9}$

Step 5.1.5

Rewrite $\sqrt{- 1 (540)}$ as $\sqrt{- 1} \cdot \sqrt{540}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{540}}{2 \cdot 9}$

Step 5.1.6

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

$x = \frac{12 \pm i \cdot \sqrt{540}}{2 \cdot 9}$

Step 5.1.7

Rewrite $540$ as $6^{2} \cdot 15$ .

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Step 5.1.7.1

Factor $36$ out of $540$ .

$x = \frac{12 \pm i \cdot \sqrt{36 (15)}}{2 \cdot 9}$

Step 5.1.7.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{12 \pm i \cdot \sqrt{6^{2} \cdot 15}}{2 \cdot 9}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (6 \sqrt{15})}{2 \cdot 9}$

Step 5.1.9

Move $6$ to the left of $i$ .

$x = \frac{12 \pm 6 i \sqrt{15}}{2 \cdot 9}$

Step 5.2

Multiply $2$ by $9$ .

$x = \frac{12 \pm 6 i \sqrt{15}}{18}$

Step 5.3

Simplify $\frac{12 \pm 6 i \sqrt{15}}{18}$ .

$x = \frac{2 \pm i \sqrt{15}}{3}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 9 \cdot 19}}{2 \cdot 9}$

Step 6.1.2

Multiply $- 4 \cdot 9 \cdot 19$ .

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Step 6.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{12 \pm \sqrt{144 - 36 \cdot 19}}{2 \cdot 9}$

Step 6.1.2.2

Multiply $- 36$ by $19$ .

$x = \frac{12 \pm \sqrt{144 - 684}}{2 \cdot 9}$

Step 6.1.3

Subtract $684$ from $144$ .

$x = \frac{12 \pm \sqrt{- 540}}{2 \cdot 9}$

Step 6.1.4

Rewrite $- 540$ as $- 1 (540)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 540}}{2 \cdot 9}$

Step 6.1.5

Rewrite $\sqrt{- 1 (540)}$ as $\sqrt{- 1} \cdot \sqrt{540}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{540}}{2 \cdot 9}$

Step 6.1.6

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

$x = \frac{12 \pm i \cdot \sqrt{540}}{2 \cdot 9}$

Step 6.1.7

Rewrite $540$ as $6^{2} \cdot 15$ .

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Step 6.1.7.1

Factor $36$ out of $540$ .

$x = \frac{12 \pm i \cdot \sqrt{36 (15)}}{2 \cdot 9}$

Step 6.1.7.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{12 \pm i \cdot \sqrt{6^{2} \cdot 15}}{2 \cdot 9}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (6 \sqrt{15})}{2 \cdot 9}$

Step 6.1.9

Move $6$ to the left of $i$ .

$x = \frac{12 \pm 6 i \sqrt{15}}{2 \cdot 9}$

Step 6.2

Multiply $2$ by $9$ .

$x = \frac{12 \pm 6 i \sqrt{15}}{18}$

Step 6.3

Simplify $\frac{12 \pm 6 i \sqrt{15}}{18}$ .

$x = \frac{2 \pm i \sqrt{15}}{3}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{2 + i \sqrt{15}}{3}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 9 \cdot 19}}{2 \cdot 9}$

Step 7.1.2

Multiply $- 4 \cdot 9 \cdot 19$ .

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Step 7.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{12 \pm \sqrt{144 - 36 \cdot 19}}{2 \cdot 9}$

Step 7.1.2.2

Multiply $- 36$ by $19$ .

$x = \frac{12 \pm \sqrt{144 - 684}}{2 \cdot 9}$

Step 7.1.3

Subtract $684$ from $144$ .

$x = \frac{12 \pm \sqrt{- 540}}{2 \cdot 9}$

Step 7.1.4

Rewrite $- 540$ as $- 1 (540)$ .

$x = \frac{12 \pm \sqrt{- 1 \cdot 540}}{2 \cdot 9}$

Step 7.1.5

Rewrite $\sqrt{- 1 (540)}$ as $\sqrt{- 1} \cdot \sqrt{540}$ .

$x = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{540}}{2 \cdot 9}$

Step 7.1.6

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

$x = \frac{12 \pm i \cdot \sqrt{540}}{2 \cdot 9}$

Step 7.1.7

Rewrite $540$ as $6^{2} \cdot 15$ .

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Step 7.1.7.1

Factor $36$ out of $540$ .

$x = \frac{12 \pm i \cdot \sqrt{36 (15)}}{2 \cdot 9}$

Step 7.1.7.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{12 \pm i \cdot \sqrt{6^{2} \cdot 15}}{2 \cdot 9}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{12 \pm i \cdot (6 \sqrt{15})}{2 \cdot 9}$

Step 7.1.9

Move $6$ to the left of $i$ .

$x = \frac{12 \pm 6 i \sqrt{15}}{2 \cdot 9}$

Step 7.2

Multiply $2$ by $9$ .

$x = \frac{12 \pm 6 i \sqrt{15}}{18}$

Step 7.3

Simplify $\frac{12 \pm 6 i \sqrt{15}}{18}$ .

$x = \frac{2 \pm i \sqrt{15}}{3}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{2 - i \sqrt{15}}{3}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{2 + i \sqrt{15}}{3}, \frac{2 - i \sqrt{15}}{3}$