Pre-Algebra Examples

$1 + (\frac{4 \sqrt{6}}{3}) = 3 x^{2} + 6 x + 8$

Step 1

Rewrite the equation as $3 x^{2} + 6 x + 8 = 1 + \frac{4 \sqrt{6}}{3}$ .

$3 x^{2} + 6 x + 8 = 1 + \frac{4 \sqrt{6}}{3}$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Move all the expressions to the left side of the equation.

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Step 2.1.1

Subtract $1$ from both sides of the equation.

$3 x^{2} + 6 x + 8 - 1 = \frac{4 \sqrt{6}}{3}$

Step 2.1.2

Subtract $\frac{4 \sqrt{6}}{3}$ from both sides of the equation.

$3 x^{2} + 6 x + 8 - 1 - \frac{4 \sqrt{6}}{3} = 0$

Step 2.2

Subtract $1$ from $8$ .

$3 x^{2} + 6 x + 7 - \frac{4 \sqrt{6}}{3} = 0$

Step 3

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.1

Apply the distributive property.

$3 (3 x^{2}) + 3 (6 x) + 3 \cdot 7 + 3 (- \frac{4 \sqrt{6}}{3}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $3$ by $3$ .

$9 x^{2} + 3 (6 x) + 3 \cdot 7 + 3 (- \frac{4 \sqrt{6}}{3}) = 0$

Step 3.2.2

Multiply $6$ by $3$ .

$9 x^{2} + 18 x + 3 \cdot 7 + 3 (- \frac{4 \sqrt{6}}{3}) = 0$

Step 3.2.3

Multiply $3$ by $7$ .

$9 x^{2} + 18 x + 21 + 3 (- \frac{4 \sqrt{6}}{3}) = 0$

Step 3.2.4

Cancel the common factor of $3$ .

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Step 3.2.4.1

Move the leading negative in $- \frac{4 \sqrt{6}}{3}$ into the numerator.

$9 x^{2} + 18 x + 21 + 3 (\frac{- 4 \sqrt{6}}{3}) = 0$

Step 3.2.4.2

Cancel the common factor.

$9 x^{2} + 18 x + 21 + 3 (\frac{- 4 \sqrt{6}}{3}) = 0$

Step 3.2.4.3

Rewrite the expression.

$9 x^{2} + 18 x + 21 - 4 \sqrt{6} = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 9$ , $b = 18$ , and $c = 21 - 4 \sqrt{6}$ into the quadratic formula and solve for $x$ .

$\frac{- 18 \pm \sqrt{18^{2} - 4 \cdot (9 \cdot (21 - 4 \sqrt{6}))}}{2 \cdot 9}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $18$ to the power of $2$ .

$x = \frac{- 18 \pm \sqrt{324 - 4 \cdot 9 \cdot (21 - 4 \sqrt{6})}}{2 \cdot 9}$

Step 6.1.2

Multiply $- 4$ by $9$ .

$x = \frac{- 18 \pm \sqrt{324 - 36 \cdot (21 - 4 \sqrt{6})}}{2 \cdot 9}$

Step 6.1.3

Apply the distributive property.

$x = \frac{- 18 \pm \sqrt{324 - 36 \cdot 21 - 36 (- 4 \sqrt{6})}}{2 \cdot 9}$

Step 6.1.4

Multiply $- 36$ by $21$ .

$x = \frac{- 18 \pm \sqrt{324 - 756 - 36 (- 4 \sqrt{6})}}{2 \cdot 9}$

Step 6.1.5

Multiply $- 4$ by $- 36$ .

$x = \frac{- 18 \pm \sqrt{324 - 756 + 144 \sqrt{6}}}{2 \cdot 9}$

Step 6.1.6

Subtract $756$ from $324$ .

$x = \frac{- 18 \pm \sqrt{- 432 + 144 \sqrt{6}}}{2 \cdot 9}$

Step 6.1.7

Rewrite $- 432 + 144 \sqrt{6}$ as $12^{2} (- 3 + \sqrt{6})$ .

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Step 6.1.7.1

Factor $144$ out of $- 432$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3) + 144 \sqrt{6}}}{2 \cdot 9}$

Step 6.1.7.2

Factor $144$ out of $144 \sqrt{6}$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3) + 144 (\sqrt{6})}}{2 \cdot 9}$

Step 6.1.7.3

Factor $144$ out of $144 (- 3) + 144 (\sqrt{6})$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3 + \sqrt{6})}}{2 \cdot 9}$

Step 6.1.7.4

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

$x = \frac{- 18 \pm \sqrt{12^{2} (- 3 + \sqrt{6})}}{2 \cdot 9}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{2 \cdot 9}$

Step 6.2

Multiply $2$ by $9$ .

$x = \frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{18}$

Step 6.3

Simplify $\frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{18}$ .

$x = \frac{- 3 \pm 2 \sqrt{- 3 + \sqrt{6}}}{3}$

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 $18$ to the power of $2$ .

$x = \frac{- 18 \pm \sqrt{324 - 4 \cdot 9 \cdot (21 - 4 \sqrt{6})}}{2 \cdot 9}$

Step 7.1.2

Multiply $- 4$ by $9$ .

$x = \frac{- 18 \pm \sqrt{324 - 36 \cdot (21 - 4 \sqrt{6})}}{2 \cdot 9}$

Step 7.1.3

Apply the distributive property.

$x = \frac{- 18 \pm \sqrt{324 - 36 \cdot 21 - 36 (- 4 \sqrt{6})}}{2 \cdot 9}$

Step 7.1.4

Multiply $- 36$ by $21$ .

$x = \frac{- 18 \pm \sqrt{324 - 756 - 36 (- 4 \sqrt{6})}}{2 \cdot 9}$

Step 7.1.5

Multiply $- 4$ by $- 36$ .

$x = \frac{- 18 \pm \sqrt{324 - 756 + 144 \sqrt{6}}}{2 \cdot 9}$

Step 7.1.6

Subtract $756$ from $324$ .

$x = \frac{- 18 \pm \sqrt{- 432 + 144 \sqrt{6}}}{2 \cdot 9}$

Step 7.1.7

Rewrite $- 432 + 144 \sqrt{6}$ as $12^{2} (- 3 + \sqrt{6})$ .

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Step 7.1.7.1

Factor $144$ out of $- 432$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3) + 144 \sqrt{6}}}{2 \cdot 9}$

Step 7.1.7.2

Factor $144$ out of $144 \sqrt{6}$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3) + 144 (\sqrt{6})}}{2 \cdot 9}$

Step 7.1.7.3

Factor $144$ out of $144 (- 3) + 144 (\sqrt{6})$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3 + \sqrt{6})}}{2 \cdot 9}$

Step 7.1.7.4

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

$x = \frac{- 18 \pm \sqrt{12^{2} (- 3 + \sqrt{6})}}{2 \cdot 9}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{2 \cdot 9}$

Step 7.2

Multiply $2$ by $9$ .

$x = \frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{18}$

Step 7.3

Simplify $\frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{18}$ .

$x = \frac{- 3 \pm 2 \sqrt{- 3 + \sqrt{6}}}{3}$

Step 7.4

Change the $\pm$ to $+$ .

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

Step 7.5

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 7.6

Factor $- 1$ out of $2 \sqrt{- 3 + \sqrt{6}}$ .

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

Step 7.7

Factor $- 1$ out of $- 1 (3) - (- 2 \sqrt{- 3 + \sqrt{6}})$ .

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

Step 7.8

Move the negative in front of the fraction.

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

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 $18$ to the power of $2$ .

$x = \frac{- 18 \pm \sqrt{324 - 4 \cdot 9 \cdot (21 - 4 \sqrt{6})}}{2 \cdot 9}$

Step 8.1.2

Multiply $- 4$ by $9$ .

$x = \frac{- 18 \pm \sqrt{324 - 36 \cdot (21 - 4 \sqrt{6})}}{2 \cdot 9}$

Step 8.1.3

Apply the distributive property.

$x = \frac{- 18 \pm \sqrt{324 - 36 \cdot 21 - 36 (- 4 \sqrt{6})}}{2 \cdot 9}$

Step 8.1.4

Multiply $- 36$ by $21$ .

$x = \frac{- 18 \pm \sqrt{324 - 756 - 36 (- 4 \sqrt{6})}}{2 \cdot 9}$

Step 8.1.5

Multiply $- 4$ by $- 36$ .

$x = \frac{- 18 \pm \sqrt{324 - 756 + 144 \sqrt{6}}}{2 \cdot 9}$

Step 8.1.6

Subtract $756$ from $324$ .

$x = \frac{- 18 \pm \sqrt{- 432 + 144 \sqrt{6}}}{2 \cdot 9}$

Step 8.1.7

Rewrite $- 432 + 144 \sqrt{6}$ as $12^{2} (- 3 + \sqrt{6})$ .

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Step 8.1.7.1

Factor $144$ out of $- 432$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3) + 144 \sqrt{6}}}{2 \cdot 9}$

Step 8.1.7.2

Factor $144$ out of $144 \sqrt{6}$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3) + 144 (\sqrt{6})}}{2 \cdot 9}$

Step 8.1.7.3

Factor $144$ out of $144 (- 3) + 144 (\sqrt{6})$ .

$x = \frac{- 18 \pm \sqrt{144 (- 3 + \sqrt{6})}}{2 \cdot 9}$

Step 8.1.7.4

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

$x = \frac{- 18 \pm \sqrt{12^{2} (- 3 + \sqrt{6})}}{2 \cdot 9}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{2 \cdot 9}$

Step 8.2

Multiply $2$ by $9$ .

$x = \frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{18}$

Step 8.3

Simplify $\frac{- 18 \pm 12 \sqrt{- 3 + \sqrt{6}}}{18}$ .

$x = \frac{- 3 \pm 2 \sqrt{- 3 + \sqrt{6}}}{3}$

Step 8.4

Change the $\pm$ to $-$ .

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

Step 8.5

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 8.6

Factor $- 1$ out of $- 2 \sqrt{- 3 + \sqrt{6}}$ .

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

Step 8.7

Factor $- 1$ out of $- 1 (3) - (2 \sqrt{- 3 + \sqrt{6}})$ .

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

Step 8.8

Move the negative in front of the fraction.

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

Step 9

The final answer is the combination of both solutions.

$x = - \frac{3 - 2 \sqrt{- 3 + \sqrt{6}}}{3}, - \frac{3 + 2 \sqrt{- 3 + \sqrt{6}}}{3}$