Calculus Examples

$3 x^{2} - 9 y = 0$ , $- 9 x + 3 y^{2} = 0$

Step 1

Solve for $y$ in $3 x^{2} - 9 y = 0$ .

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Step 1.1

Subtract $3 x^{2}$ from both sides of the equation.

$- 9 y = - 3 x^{2}$

$- 9 x + 3 y^{2} = 0$

Step 1.2

Divide each term in $- 9 y = - 3 x^{2}$ by $- 9$ and simplify.

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Step 1.2.1

Divide each term in $- 9 y = - 3 x^{2}$ by $- 9$ .

$\frac{- 9 y}{- 9} = \frac{- 3 x^{2}}{- 9}$

$- 9 x + 3 y^{2} = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 9 y}{- 9} = \frac{- 3 x^{2}}{- 9}$

$- 9 x + 3 y^{2} = 0$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 x^{2}}{- 9}$

$- 9 x + 3 y^{2} = 0$

$y = \frac{- 3 x^{2}}{- 9}$

$- 9 x + 3 y^{2} = 0$

$y = \frac{- 3 x^{2}}{- 9}$

$- 9 x + 3 y^{2} = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $- 3$ and $- 9$ .

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Step 1.2.3.1.1

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

$y = \frac{- 3 x^{2}}{- 9}$

$- 9 x + 3 y^{2} = 0$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $- 3$ out of $- 9$ .

$y = \frac{- 3 x^{2}}{- 3 \cdot 3}$

$- 9 x + 3 y^{2} = 0$

Step 1.2.3.1.2.2

Cancel the common factor.

$y = \frac{- 3 x^{2}}{- 3 \cdot 3}$

$- 9 x + 3 y^{2} = 0$

Step 1.2.3.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{3}$

$- 9 x + 3 y^{2} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + 3 y^{2} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + 3 y^{2} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + 3 y^{2} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + 3 y^{2} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + 3 y^{2} = 0$

Step 2

Replace all occurrences of $y$ with $\frac{x^{2}}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $- 9 x + 3 y^{2} = 0$ with $\frac{x^{2}}{3}$ .

$- 9 x + 3 {(\frac{x^{2}}{3})}^{2} = 0$

$y = \frac{x^{2}}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the product rule to $\frac{x^{2}}{3}$ .

$- 9 x + 3 (\frac{{(x^{2})}^{2}}{3^{2}}) = 0$

$y = \frac{x^{2}}{3}$

Step 2.2.1.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

$y = \frac{x^{2}}{3}$

Step 2.2.1.2.2

Multiply $2$ by $2$ .

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

$y = \frac{x^{2}}{3}$

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

$y = \frac{x^{2}}{3}$

Step 2.2.1.3

Raise $3$ to the power of $2$ .

$- 9 x + 3 (\frac{x^{4}}{9}) = 0$

$y = \frac{x^{2}}{3}$

Step 2.2.1.4

Cancel the common factor of $3$ .

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Step 2.2.1.4.1

Factor $3$ out of $9$ .

$- 9 x + 3 (\frac{x^{4}}{3 (3)}) = 0$

$y = \frac{x^{2}}{3}$

Step 2.2.1.4.2

Cancel the common factor.

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

$y = \frac{x^{2}}{3}$

Step 2.2.1.4.3

Rewrite the expression.

$- 9 x + \frac{x^{4}}{3} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + \frac{x^{4}}{3} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + \frac{x^{4}}{3} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + \frac{x^{4}}{3} = 0$

$y = \frac{x^{2}}{3}$

$- 9 x + \frac{x^{4}}{3} = 0$

$y = \frac{x^{2}}{3}$

Step 3

Solve for $x$ in $- 9 x + \frac{x^{4}}{3} = 0$ .

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Step 3.1

Factor $- x$ out of $- 9 x + \frac{x^{4}}{3}$ .

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Step 3.1.1

Reorder $- 9 x$ and $\frac{x^{4}}{3}$ .

$\frac{x^{4}}{3} - 9 x = 0$

$y = \frac{x^{2}}{3}$

Step 3.1.2

Factor $- x$ out of $\frac{x^{4}}{3}$ .

$- x \frac{- x^{3}}{3} - 9 x = 0$

$y = \frac{x^{2}}{3}$

Step 3.1.3

Factor $- x$ out of $- 9 x$ .

$- x \frac{- x^{3}}{3} - x \cdot 9 = 0$

$y = \frac{x^{2}}{3}$

Step 3.1.4

Factor $- x$ out of $- x \frac{- x^{3}}{3} - x (9)$ .

$- x (\frac{- x^{3}}{3} + 9) = 0$

$y = \frac{x^{2}}{3}$

$- x (\frac{- x^{3}}{3} + 9) = 0$

$y = \frac{x^{2}}{3}$

Step 3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$\frac{- x^{3}}{3} + 9 = 0$

$y = \frac{x^{2}}{3}$

Step 3.3

Set $x$ equal to $0$ .

$x = 0$

$y = \frac{x^{2}}{3}$

Step 3.4

Set $\frac{- x^{3}}{3} + 9$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $\frac{- x^{3}}{3} + 9$ equal to $0$ .

$\frac{- x^{3}}{3} + 9 = 0$

$y = \frac{x^{2}}{3}$

Step 3.4.2

Solve $\frac{- x^{3}}{3} + 9 = 0$ for $x$ .

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Step 3.4.2.1

Move the negative in front of the fraction.

$- \frac{x^{3}}{3} + 9 = 0$

$y = \frac{x^{2}}{3}$

Step 3.4.2.2

Subtract $9$ from both sides of the equation.

$- \frac{x^{3}}{3} = - 9$

$y = \frac{x^{2}}{3}$

Step 3.4.2.3

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{x^{3}}{3}) = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

Step 3.4.2.4

Simplify both sides of the equation.

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Step 3.4.2.4.1

Simplify the left side.

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Step 3.4.2.4.1.1

Simplify $- 3 (- \frac{x^{3}}{3})$ .

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Step 3.4.2.4.1.1.1

Cancel the common factor of $3$ .

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Step 3.4.2.4.1.1.1.1

Move the leading negative in $- \frac{x^{3}}{3}$ into the numerator.

$- 3 \frac{- x^{3}}{3} = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

Step 3.4.2.4.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- x^{3}}{3}) = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

Step 3.4.2.4.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- x^{3}}{3}) = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

Step 3.4.2.4.1.1.1.4

Rewrite the expression.

$x^{3} = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

$x^{3} = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

Step 3.4.2.4.1.1.2

Multiply.

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Step 3.4.2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x^{3} = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

Step 3.4.2.4.1.1.2.2

Multiply $x^{3}$ by $1$ .

$x^{3} = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

$x^{3} = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

$x^{3} = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

$x^{3} = - 3 \cdot - 9$

$y = \frac{x^{2}}{3}$

Step 3.4.2.4.2

Simplify the right side.

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Step 3.4.2.4.2.1

Multiply $- 3$ by $- 9$ .

$x^{3} = 27$

$y = \frac{x^{2}}{3}$

$x^{3} = 27$

$y = \frac{x^{2}}{3}$

$x^{3} = 27$

$y = \frac{x^{2}}{3}$

Step 3.4.2.5

Subtract $27$ from both sides of the equation.

$x^{3} - 27 = 0$

$y = \frac{x^{2}}{3}$

Step 3.4.2.6

Factor the left side of the equation.

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Step 3.4.2.6.1

Rewrite $27$ as $3^{3}$ .

$x^{3} - 3^{3} = 0$

$y = \frac{x^{2}}{3}$

Step 3.4.2.6.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 3$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.6.3

Simplify.

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Step 3.4.2.6.3.1

Move $3$ to the left of $x$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.6.3.2

Multiply $3$ by $x$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.6.3.3

Raise $3$ to the power of $2$ .

$(x - 3) (x^{2} + 3 x + 9) = 0$

$y = \frac{x^{2}}{3}$

$(x - 3) (x^{2} + 3 x + 9) = 0$

$y = \frac{x^{2}}{3}$

$(x - 3) (x^{2} + 3 x + 9) = 0$

$y = \frac{x^{2}}{3}$

Step 3.4.2.7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x^{2} + 3 x + 9 = 0$

$y = \frac{x^{2}}{3}$

Step 3.4.2.8

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 3.4.2.8.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

$y = \frac{x^{2}}{3}$

Step 3.4.2.8.2

Add $3$ to both sides of the equation.

$x = 3$

$y = \frac{x^{2}}{3}$

$x = 3$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9

Set $x^{2} + 3 x + 9$ equal to $0$ and solve for $x$ .

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Step 3.4.2.9.1

Set $x^{2} + 3 x + 9$ equal to $0$ .

$x^{2} + 3 x + 9 = 0$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2

Solve $x^{2} + 3 x + 9 = 0$ for $x$ .

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Step 3.4.2.9.2.1

Use the quadratic formula to find the solutions.

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3

Simplify.

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Step 3.4.2.9.2.3.1

Simplify the numerator.

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Step 3.4.2.9.2.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 3.4.2.9.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.3

Subtract $36$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.6

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

$x = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.4.2.9.2.3.1.7.1

Factor $9$ out of $27$ .

$x = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.3.2

Multiply $2$ by $1$ .

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

$y = \frac{x^{2}}{3}$

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4

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

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Step 3.4.2.9.2.4.1

Simplify the numerator.

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Step 3.4.2.9.2.4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 3.4.2.9.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.3

Subtract $36$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.6

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

$x = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.4.2.9.2.4.1.7.1

Factor $9$ out of $27$ .

$x = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.2

Multiply $2$ by $1$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.3

Change the $\pm$ to $+$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.4

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

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.5

Factor $- 1$ out of $3 i \sqrt{3}$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.6

Factor $- 1$ out of $- 1 (3) - (- 3 i \sqrt{3})$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.4.7

Move the negative in front of the fraction.

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

$y = \frac{x^{2}}{3}$

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5

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

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Step 3.4.2.9.2.5.1

Simplify the numerator.

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Step 3.4.2.9.2.5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 3.4.2.9.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.3

Subtract $36$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.6

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

$x = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.4.2.9.2.5.1.7.1

Factor $9$ out of $27$ .

$x = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.2

Multiply $2$ by $1$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.3

Change the $\pm$ to $-$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.4

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

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.5

Factor $- 1$ out of $- 3 i \sqrt{3}$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.6

Factor $- 1$ out of $- 1 (3) - (3 i \sqrt{3})$ .

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.5.7

Move the negative in front of the fraction.

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

$y = \frac{x^{2}}{3}$

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.9.2.6

The final answer is the combination of both solutions.

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

$y = \frac{x^{2}}{3}$

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

$y = \frac{x^{2}}{3}$

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

$y = \frac{x^{2}}{3}$

Step 3.4.2.10

The final solution is all the values that make $(x - 3) (x^{2} + 3 x + 9) = 0$ true.

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

$y = \frac{x^{2}}{3}$

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

$y = \frac{x^{2}}{3}$

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

$y = \frac{x^{2}}{3}$

Step 3.5

The final solution is all the values that make $- x (\frac{- x^{3}}{3} + 9) = 0$ true.

$x = 0, 3, - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

$y = \frac{x^{2}}{3}$

$x = 0, 3, - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

$y = \frac{x^{2}}{3}$

Step 4

Replace all occurrences of $x$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $0$ .

$y = \frac{{(0)}^{2}}{3}$

$x = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{{(0)}^{2}}{3}$ .

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0}{3}$

$x = 0$

Step 4.2.1.2

Divide $0$ by $3$ .

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

Step 5

Replace all occurrences of $x$ with $3$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $3$ .

$y = \frac{{(3)}^{2}}{3}$

$x = 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Cancel the common factor of ${(3)}^{2}$ and $3$ .

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Step 5.2.1.1

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

$y = \frac{3 \cdot 3}{3}$

$x = 3$

Step 5.2.1.2

Cancel the common factors.

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Step 5.2.1.2.1

Factor $3$ out of $3$ .

$y = \frac{3 \cdot 3}{3 (1)}$

$x = 3$

Step 5.2.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 3}{3 \cdot 1}$

$x = 3$

Step 5.2.1.2.3

Rewrite the expression.

$y = \frac{3}{1}$

$x = 3$

Step 5.2.1.2.4

Divide $3$ by $1$ .

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

Step 6

Replace all occurrences of $x$ with $0$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $0$ .

$y = \frac{{(0)}^{2}}{3}$

$x = 0$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{{(0)}^{2}}{3}$ .

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Step 6.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0}{3}$

$x = 0$

Step 6.2.1.2

Divide $0$ by $3$ .

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

Step 7

Replace all occurrences of $x$ with $3$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $3$ .

$y = \frac{{(3)}^{2}}{3}$

$x = 3$

Step 7.2

Simplify the right side.

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Step 7.2.1

Cancel the common factor of ${(3)}^{2}$ and $3$ .

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Step 7.2.1.1

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

$y = \frac{3 \cdot 3}{3}$

$x = 3$

Step 7.2.1.2

Cancel the common factors.

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Step 7.2.1.2.1

Factor $3$ out of $3$ .

$y = \frac{3 \cdot 3}{3 (1)}$

$x = 3$

Step 7.2.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 3}{3 \cdot 1}$

$x = 3$

Step 7.2.1.2.3

Rewrite the expression.

$y = \frac{3}{1}$

$x = 3$

Step 7.2.1.2.4

Divide $3$ by $1$ .

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

Step 8

Replace all occurrences of $x$ with $- \frac{3 - 3 i \sqrt{3}}{2}$ in each equation.

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Step 8.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $- \frac{3 - 3 i \sqrt{3}}{2}$ .

$y = \frac{{(- \frac{3 - 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{{(- \frac{3 - 3 i \sqrt{3}}{2})}^{2}}{3}$ .

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Step 8.2.1.1

Simplify the numerator.

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Step 8.2.1.1.1

Apply the product rule to $- \frac{3 - 3 i \sqrt{3}}{2}$ .

$y = \frac{{(- 1)}^{2} {(\frac{3 - 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 8.2.1.1.2

Raise $- 1$ to the power of $2$ .

$y = \frac{1 {(\frac{3 - 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 8.2.1.1.3

Apply the product rule to $\frac{3 - 3 i \sqrt{3}}{2}$ .

$y = \frac{1 (\frac{{(3 - 3 i \sqrt{3})}^{2}}{2^{2}})}{3}$

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

Step 8.2.1.1.4

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

$y = \frac{1 (\frac{{(3 - 3 i \sqrt{3})}^{2}}{4})}{3}$

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

Step 8.2.1.1.5

Rewrite ${(3 - 3 i \sqrt{3})}^{2}$ as $(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})$ .

$y = \frac{1 (\frac{(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.6

Expand $(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})$ using the FOIL Method.

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Step 8.2.1.1.6.1

Apply the distributive property.

$y = \frac{1 (\frac{3 (3 - 3 i \sqrt{3}) - 3 i \sqrt{3} (3 - 3 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.6.2

Apply the distributive property.

$y = \frac{1 (\frac{3 \cdot 3 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} (3 - 3 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.6.3

Apply the distributive property.

$y = \frac{1 (\frac{3 \cdot 3 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

$y = \frac{1 (\frac{3 \cdot 3 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.7

Simplify and combine like terms.

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Step 8.2.1.1.7.1

Simplify each term.

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Step 8.2.1.1.7.1.1

Multiply $3$ by $3$ .

$y = \frac{1 (\frac{9 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.7.1.2

Multiply $- 3$ by $3$ .

$y = \frac{1 (\frac{9 - 9 (i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.7.1.3

Multiply $3$ by $- 3$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.7.1.4

Multiply $- 3 i \sqrt{3} (- 3 i \sqrt{3})$ .

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Step 8.2.1.1.7.1.4.1

Multiply $- 3$ by $- 3$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i \sqrt{3} (i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.7.1.4.2

Raise $i$ to the power of $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 (i i) \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 8.2.1.1.7.1.4.3

Raise $i$ to the power of $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 (i i) \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 8.2.1.1.7.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{1 + 1} \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 8.2.1.1.7.1.4.5

Add $1$ and $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 8.2.1.1.7.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} (\sqrt{3} \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.7.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} (\sqrt{3} \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.7.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{1 + 1}}{4})}{3}$

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

Step 8.2.1.1.7.1.4.9

Add $1$ and $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4})}{3}$

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4})}{3}$

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

Step 8.2.1.1.7.1.5

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 \cdot (- 1 {\sqrt{3}}^{2})}{4})}{3}$

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

Step 8.2.1.1.7.1.6

Multiply $9$ by $- 1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 {\sqrt{3}}^{2}}{4})}{3}$

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

Step 8.2.1.1.7.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.2.1.1.7.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 {(3^{\frac{1}{2}})}^{2}}{4})}{3}$

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

Step 8.2.1.1.7.1.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{1}{2} \cdot 2}}{4})}{3}$

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

Step 8.2.1.1.7.1.7.3

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4})}{3}$

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

Step 8.2.1.1.7.1.7.4

Cancel the common factor of $2$ .

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Step 8.2.1.1.7.1.7.4.1

Cancel the common factor.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4})}{3}$

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

Step 8.2.1.1.7.1.7.4.2

Rewrite the expression.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

Step 8.2.1.1.7.1.7.5

Evaluate the exponent.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

Step 8.2.1.1.7.1.8

Multiply $- 9$ by $3$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 27}{4})}{3}$

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 27}{4})}{3}$

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

Step 8.2.1.1.7.2

Subtract $27$ from $9$ .

$y = \frac{1 (\frac{- 9 i \sqrt{3} - 9 i \sqrt{3} - 18}{4})}{3}$

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

Step 8.2.1.1.7.3

Subtract $9 i \sqrt{3}$ from $- 9 i \sqrt{3}$ .

$y = \frac{1 (\frac{- 18 i \sqrt{3} - 18}{4})}{3}$

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

$y = \frac{1 (\frac{- 18 i \sqrt{3} - 18}{4})}{3}$

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

Step 8.2.1.1.8

Reorder $- 18 i \sqrt{3}$ and $- 18$ .

$y = \frac{1 (\frac{- 18 - 18 i \sqrt{3}}{4})}{3}$

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

Step 8.2.1.1.9

Cancel the common factor of $- 18 - 18 i \sqrt{3}$ and $4$ .

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Step 8.2.1.1.9.1

Factor $2$ out of $- 18$ .

$y = \frac{1 (\frac{2 (- 9) - 18 i \sqrt{3}}{4})}{3}$

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

Step 8.2.1.1.9.2

Factor $2$ out of $- 18 i \sqrt{3}$ .

$y = \frac{1 (\frac{2 (- 9) + 2 (- 9 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.9.3

Factor $2$ out of $2 (- 9) + 2 (- 9 i \sqrt{3})$ .

$y = \frac{1 (\frac{2 (- 9 - 9 i \sqrt{3})}{4})}{3}$

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

Step 8.2.1.1.9.4

Cancel the common factors.

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Step 8.2.1.1.9.4.1

Factor $2$ out of $4$ .

$y = \frac{1 (\frac{2 (- 9 - 9 i \sqrt{3})}{2 \cdot 2})}{3}$

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

Step 8.2.1.1.9.4.2

Cancel the common factor.

$y = \frac{1 (\frac{2 (- 9 - 9 i \sqrt{3})}{2 \cdot 2})}{3}$

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

Step 8.2.1.1.9.4.3

Rewrite the expression.

$y = \frac{1 (\frac{- 9 - 9 i \sqrt{3}}{2})}{3}$

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

$y = \frac{1 (\frac{- 9 - 9 i \sqrt{3}}{2})}{3}$

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

$y = \frac{1 (\frac{- 9 - 9 i \sqrt{3}}{2})}{3}$

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

Step 8.2.1.1.10

Multiply $\frac{- 9 - 9 i \sqrt{3}}{2}$ by $1$ .

$y = \frac{\frac{- 9 - 9 i \sqrt{3}}{2}}{3}$

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

$y = \frac{\frac{- 9 - 9 i \sqrt{3}}{2}}{3}$

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

Step 8.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{- 9 - 9 i \sqrt{3}}{2} \cdot \frac{1}{3}$

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

Step 8.2.1.3

Multiply $\frac{- 9 - 9 i \sqrt{3}}{2} \cdot \frac{1}{3}$ .

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Step 8.2.1.3.1

Multiply $\frac{- 9 - 9 i \sqrt{3}}{2}$ by $\frac{1}{3}$ .

$y = \frac{- 9 - 9 i \sqrt{3}}{2 \cdot 3}$

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

Step 8.2.1.3.2

Multiply $2$ by $3$ .

$y = \frac{- 9 - 9 i \sqrt{3}}{6}$

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

$y = \frac{- 9 - 9 i \sqrt{3}}{6}$

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

Step 8.2.1.4

Cancel the common factor of $- 9 - 9 i \sqrt{3}$ and $6$ .

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Step 8.2.1.4.1

Factor $3$ out of $- 9$ .

$y = \frac{3 (- 3) - 9 i \sqrt{3}}{6}$

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

Step 8.2.1.4.2

Factor $3$ out of $- 9 i \sqrt{3}$ .

$y = \frac{3 (- 3) + 3 (- 3 i \sqrt{3})}{6}$

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

Step 8.2.1.4.3

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

$y = \frac{3 (- 3 - 3 i \sqrt{3})}{6}$

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

Step 8.2.1.4.4

Cancel the common factors.

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Step 8.2.1.4.4.1

Factor $3$ out of $6$ .

$y = \frac{3 (- 3 - 3 i \sqrt{3})}{3 \cdot 2}$

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

Step 8.2.1.4.4.2

Cancel the common factor.

$y = \frac{3 (- 3 - 3 i \sqrt{3})}{3 \cdot 2}$

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

Step 8.2.1.4.4.3

Rewrite the expression.

$y = \frac{- 3 - 3 i \sqrt{3}}{2}$

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

$y = \frac{- 3 - 3 i \sqrt{3}}{2}$

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

$y = \frac{- 3 - 3 i \sqrt{3}}{2}$

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

Step 8.2.1.5

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

$y = \frac{- 1 \cdot 3 - 3 i \sqrt{3}}{2}$

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

Step 8.2.1.6

Factor $- 1$ out of $- 3 i \sqrt{3}$ .

$y = \frac{- 1 \cdot 3 - (3 i \sqrt{3})}{2}$

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

Step 8.2.1.7

Factor $- 1$ out of $- 1 (3) - (3 i \sqrt{3})$ .

$y = \frac{- 1 (3 + 3 i \sqrt{3})}{2}$

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

Step 8.2.1.8

Move the negative in front of the fraction.

$y = - \frac{3 + 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 + 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 + 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 + 3 i \sqrt{3}}{2}$

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

Step 9

Replace all occurrences of $x$ with $0$ in each equation.

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Step 9.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $0$ .

$y = \frac{{(0)}^{2}}{3}$

$x = 0$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $\frac{{(0)}^{2}}{3}$ .

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Step 9.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0}{3}$

$x = 0$

Step 9.2.1.2

Divide $0$ by $3$ .

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

Step 10

Replace all occurrences of $x$ with $3$ in each equation.

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Step 10.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $3$ .

$y = \frac{{(3)}^{2}}{3}$

$x = 3$

Step 10.2

Simplify the right side.

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Step 10.2.1

Cancel the common factor of ${(3)}^{2}$ and $3$ .

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Step 10.2.1.1

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

$y = \frac{3 \cdot 3}{3}$

$x = 3$

Step 10.2.1.2

Cancel the common factors.

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Step 10.2.1.2.1

Factor $3$ out of $3$ .

$y = \frac{3 \cdot 3}{3 (1)}$

$x = 3$

Step 10.2.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 3}{3 \cdot 1}$

$x = 3$

Step 10.2.1.2.3

Rewrite the expression.

$y = \frac{3}{1}$

$x = 3$

Step 10.2.1.2.4

Divide $3$ by $1$ .

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

Step 11

Replace all occurrences of $x$ with $- \frac{3 - 3 i \sqrt{3}}{2}$ in each equation.

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Step 11.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $- \frac{3 - 3 i \sqrt{3}}{2}$ .

$y = \frac{{(- \frac{3 - 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $\frac{{(- \frac{3 - 3 i \sqrt{3}}{2})}^{2}}{3}$ .

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Step 11.2.1.1

Simplify the numerator.

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Step 11.2.1.1.1

Apply the product rule to $- \frac{3 - 3 i \sqrt{3}}{2}$ .

$y = \frac{{(- 1)}^{2} {(\frac{3 - 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 11.2.1.1.2

Raise $- 1$ to the power of $2$ .

$y = \frac{1 {(\frac{3 - 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 11.2.1.1.3

Apply the product rule to $\frac{3 - 3 i \sqrt{3}}{2}$ .

$y = \frac{1 (\frac{{(3 - 3 i \sqrt{3})}^{2}}{2^{2}})}{3}$

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

Step 11.2.1.1.4

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

$y = \frac{1 (\frac{{(3 - 3 i \sqrt{3})}^{2}}{4})}{3}$

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

Step 11.2.1.1.5

Rewrite ${(3 - 3 i \sqrt{3})}^{2}$ as $(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})$ .

$y = \frac{1 (\frac{(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.6

Expand $(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})$ using the FOIL Method.

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Step 11.2.1.1.6.1

Apply the distributive property.

$y = \frac{1 (\frac{3 (3 - 3 i \sqrt{3}) - 3 i \sqrt{3} (3 - 3 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.6.2

Apply the distributive property.

$y = \frac{1 (\frac{3 \cdot 3 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} (3 - 3 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.6.3

Apply the distributive property.

$y = \frac{1 (\frac{3 \cdot 3 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

$y = \frac{1 (\frac{3 \cdot 3 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.7

Simplify and combine like terms.

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Step 11.2.1.1.7.1

Simplify each term.

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Step 11.2.1.1.7.1.1

Multiply $3$ by $3$ .

$y = \frac{1 (\frac{9 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.7.1.2

Multiply $- 3$ by $3$ .

$y = \frac{1 (\frac{9 - 9 (i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.7.1.3

Multiply $3$ by $- 3$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.7.1.4

Multiply $- 3 i \sqrt{3} (- 3 i \sqrt{3})$ .

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Step 11.2.1.1.7.1.4.1

Multiply $- 3$ by $- 3$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i \sqrt{3} (i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.7.1.4.2

Raise $i$ to the power of $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 (i i) \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 11.2.1.1.7.1.4.3

Raise $i$ to the power of $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 (i i) \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 11.2.1.1.7.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{1 + 1} \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 11.2.1.1.7.1.4.5

Add $1$ and $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 11.2.1.1.7.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} (\sqrt{3} \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.7.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} (\sqrt{3} \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.7.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{1 + 1}}{4})}{3}$

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

Step 11.2.1.1.7.1.4.9

Add $1$ and $1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4})}{3}$

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4})}{3}$

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

Step 11.2.1.1.7.1.5

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 \cdot (- 1 {\sqrt{3}}^{2})}{4})}{3}$

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

Step 11.2.1.1.7.1.6

Multiply $9$ by $- 1$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 {\sqrt{3}}^{2}}{4})}{3}$

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

Step 11.2.1.1.7.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 11.2.1.1.7.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 {(3^{\frac{1}{2}})}^{2}}{4})}{3}$

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

Step 11.2.1.1.7.1.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{1}{2} \cdot 2}}{4})}{3}$

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

Step 11.2.1.1.7.1.7.3

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4})}{3}$

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

Step 11.2.1.1.7.1.7.4

Cancel the common factor of $2$ .

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Step 11.2.1.1.7.1.7.4.1

Cancel the common factor.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4})}{3}$

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

Step 11.2.1.1.7.1.7.4.2

Rewrite the expression.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

Step 11.2.1.1.7.1.7.5

Evaluate the exponent.

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

Step 11.2.1.1.7.1.8

Multiply $- 9$ by $3$ .

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 27}{4})}{3}$

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

$y = \frac{1 (\frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 27}{4})}{3}$

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

Step 11.2.1.1.7.2

Subtract $27$ from $9$ .

$y = \frac{1 (\frac{- 9 i \sqrt{3} - 9 i \sqrt{3} - 18}{4})}{3}$

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

Step 11.2.1.1.7.3

Subtract $9 i \sqrt{3}$ from $- 9 i \sqrt{3}$ .

$y = \frac{1 (\frac{- 18 i \sqrt{3} - 18}{4})}{3}$

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

$y = \frac{1 (\frac{- 18 i \sqrt{3} - 18}{4})}{3}$

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

Step 11.2.1.1.8

Reorder $- 18 i \sqrt{3}$ and $- 18$ .

$y = \frac{1 (\frac{- 18 - 18 i \sqrt{3}}{4})}{3}$

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

Step 11.2.1.1.9

Cancel the common factor of $- 18 - 18 i \sqrt{3}$ and $4$ .

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Step 11.2.1.1.9.1

Factor $2$ out of $- 18$ .

$y = \frac{1 (\frac{2 (- 9) - 18 i \sqrt{3}}{4})}{3}$

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

Step 11.2.1.1.9.2

Factor $2$ out of $- 18 i \sqrt{3}$ .

$y = \frac{1 (\frac{2 (- 9) + 2 (- 9 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.9.3

Factor $2$ out of $2 (- 9) + 2 (- 9 i \sqrt{3})$ .

$y = \frac{1 (\frac{2 (- 9 - 9 i \sqrt{3})}{4})}{3}$

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

Step 11.2.1.1.9.4

Cancel the common factors.

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Step 11.2.1.1.9.4.1

Factor $2$ out of $4$ .

$y = \frac{1 (\frac{2 (- 9 - 9 i \sqrt{3})}{2 \cdot 2})}{3}$

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

Step 11.2.1.1.9.4.2

Cancel the common factor.

$y = \frac{1 (\frac{2 (- 9 - 9 i \sqrt{3})}{2 \cdot 2})}{3}$

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

Step 11.2.1.1.9.4.3

Rewrite the expression.

$y = \frac{1 (\frac{- 9 - 9 i \sqrt{3}}{2})}{3}$

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

$y = \frac{1 (\frac{- 9 - 9 i \sqrt{3}}{2})}{3}$

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

$y = \frac{1 (\frac{- 9 - 9 i \sqrt{3}}{2})}{3}$

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

Step 11.2.1.1.10

Multiply $\frac{- 9 - 9 i \sqrt{3}}{2}$ by $1$ .

$y = \frac{\frac{- 9 - 9 i \sqrt{3}}{2}}{3}$

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

$y = \frac{\frac{- 9 - 9 i \sqrt{3}}{2}}{3}$

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

Step 11.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{- 9 - 9 i \sqrt{3}}{2} \cdot \frac{1}{3}$

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

Step 11.2.1.3

Multiply $\frac{- 9 - 9 i \sqrt{3}}{2} \cdot \frac{1}{3}$ .

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Step 11.2.1.3.1

Multiply $\frac{- 9 - 9 i \sqrt{3}}{2}$ by $\frac{1}{3}$ .

$y = \frac{- 9 - 9 i \sqrt{3}}{2 \cdot 3}$

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

Step 11.2.1.3.2

Multiply $2$ by $3$ .

$y = \frac{- 9 - 9 i \sqrt{3}}{6}$

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

$y = \frac{- 9 - 9 i \sqrt{3}}{6}$

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

Step 11.2.1.4

Cancel the common factor of $- 9 - 9 i \sqrt{3}$ and $6$ .

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Step 11.2.1.4.1

Factor $3$ out of $- 9$ .

$y = \frac{3 (- 3) - 9 i \sqrt{3}}{6}$

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

Step 11.2.1.4.2

Factor $3$ out of $- 9 i \sqrt{3}$ .

$y = \frac{3 (- 3) + 3 (- 3 i \sqrt{3})}{6}$

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

Step 11.2.1.4.3

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

$y = \frac{3 (- 3 - 3 i \sqrt{3})}{6}$

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

Step 11.2.1.4.4

Cancel the common factors.

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Step 11.2.1.4.4.1

Factor $3$ out of $6$ .

$y = \frac{3 (- 3 - 3 i \sqrt{3})}{3 \cdot 2}$

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

Step 11.2.1.4.4.2

Cancel the common factor.

$y = \frac{3 (- 3 - 3 i \sqrt{3})}{3 \cdot 2}$

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

Step 11.2.1.4.4.3

Rewrite the expression.

$y = \frac{- 3 - 3 i \sqrt{3}}{2}$

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

$y = \frac{- 3 - 3 i \sqrt{3}}{2}$

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

$y = \frac{- 3 - 3 i \sqrt{3}}{2}$

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

Step 11.2.1.5

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

$y = \frac{- 1 \cdot 3 - 3 i \sqrt{3}}{2}$

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

Step 11.2.1.6

Factor $- 1$ out of $- 3 i \sqrt{3}$ .

$y = \frac{- 1 \cdot 3 - (3 i \sqrt{3})}{2}$

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

Step 11.2.1.7

Factor $- 1$ out of $- 1 (3) - (3 i \sqrt{3})$ .

$y = \frac{- 1 (3 + 3 i \sqrt{3})}{2}$

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

Step 11.2.1.8

Move the negative in front of the fraction.

$y = - \frac{3 + 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 + 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 + 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 + 3 i \sqrt{3}}{2}$

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

Step 12

Replace all occurrences of $x$ with $- \frac{3 + 3 i \sqrt{3}}{2}$ in each equation.

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Step 12.1

Replace all occurrences of $x$ in $y = \frac{x^{2}}{3}$ with $- \frac{3 + 3 i \sqrt{3}}{2}$ .

$y = \frac{{(- \frac{3 + 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $\frac{{(- \frac{3 + 3 i \sqrt{3}}{2})}^{2}}{3}$ .

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Step 12.2.1.1

Simplify the numerator.

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Step 12.2.1.1.1

Apply the product rule to $- \frac{3 + 3 i \sqrt{3}}{2}$ .

$y = \frac{{(- 1)}^{2} {(\frac{3 + 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 12.2.1.1.2

Raise $- 1$ to the power of $2$ .

$y = \frac{1 {(\frac{3 + 3 i \sqrt{3}}{2})}^{2}}{3}$

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

Step 12.2.1.1.3

Apply the product rule to $\frac{3 + 3 i \sqrt{3}}{2}$ .

$y = \frac{1 (\frac{{(3 + 3 i \sqrt{3})}^{2}}{2^{2}})}{3}$

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

Step 12.2.1.1.4

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

$y = \frac{1 (\frac{{(3 + 3 i \sqrt{3})}^{2}}{4})}{3}$

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

Step 12.2.1.1.5

Rewrite ${(3 + 3 i \sqrt{3})}^{2}$ as $(3 + 3 i \sqrt{3}) (3 + 3 i \sqrt{3})$ .

$y = \frac{1 (\frac{(3 + 3 i \sqrt{3}) (3 + 3 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.6

Expand $(3 + 3 i \sqrt{3}) (3 + 3 i \sqrt{3})$ using the FOIL Method.

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Step 12.2.1.1.6.1

Apply the distributive property.

$y = \frac{1 (\frac{3 (3 + 3 i \sqrt{3}) + 3 i \sqrt{3} (3 + 3 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.6.2

Apply the distributive property.

$y = \frac{1 (\frac{3 \cdot 3 + 3 (3 i \sqrt{3}) + 3 i \sqrt{3} (3 + 3 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.6.3

Apply the distributive property.

$y = \frac{1 (\frac{3 \cdot 3 + 3 (3 i \sqrt{3}) + 3 i \sqrt{3} \cdot 3 + 3 i \sqrt{3} (3 i \sqrt{3})}{4})}{3}$

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

$y = \frac{1 (\frac{3 \cdot 3 + 3 (3 i \sqrt{3}) + 3 i \sqrt{3} \cdot 3 + 3 i \sqrt{3} (3 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.7

Simplify and combine like terms.

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Step 12.2.1.1.7.1

Simplify each term.

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Step 12.2.1.1.7.1.1

Multiply $3$ by $3$ .

$y = \frac{1 (\frac{9 + 3 (3 i \sqrt{3}) + 3 i \sqrt{3} \cdot 3 + 3 i \sqrt{3} (3 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.7.1.2

Multiply $3$ by $3$ .

$y = \frac{1 (\frac{9 + 9 (i \sqrt{3}) + 3 i \sqrt{3} \cdot 3 + 3 i \sqrt{3} (3 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.7.1.3

Multiply $3$ by $3$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 3 i \sqrt{3} (3 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.7.1.4

Multiply $3 i \sqrt{3} (3 i \sqrt{3})$ .

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Step 12.2.1.1.7.1.4.1

Multiply $3$ by $3$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i \sqrt{3} (i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.7.1.4.2

Raise $i$ to the power of $1$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 (i i) \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 12.2.1.1.7.1.4.3

Raise $i$ to the power of $1$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 (i i) \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 12.2.1.1.7.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{1 + 1} \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 12.2.1.1.7.1.4.5

Add $1$ and $1$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} \sqrt{3} \sqrt{3}}{4})}{3}$

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

Step 12.2.1.1.7.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} (\sqrt{3} \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.7.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} (\sqrt{3} \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.7.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{1 + 1}}{4})}{3}$

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

Step 12.2.1.1.7.1.4.9

Add $1$ and $1$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4})}{3}$

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

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4})}{3}$

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

Step 12.2.1.1.7.1.5

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 \cdot (- 1 {\sqrt{3}}^{2})}{4})}{3}$

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

Step 12.2.1.1.7.1.6

Multiply $9$ by $- 1$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 {\sqrt{3}}^{2}}{4})}{3}$

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

Step 12.2.1.1.7.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 12.2.1.1.7.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 {(3^{\frac{1}{2}})}^{2}}{4})}{3}$

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

Step 12.2.1.1.7.1.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3^{\frac{1}{2} \cdot 2}}{4})}{3}$

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

Step 12.2.1.1.7.1.7.3

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

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4})}{3}$

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

Step 12.2.1.1.7.1.7.4

Cancel the common factor of $2$ .

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Step 12.2.1.1.7.1.7.4.1

Cancel the common factor.

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4})}{3}$

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

Step 12.2.1.1.7.1.7.4.2

Rewrite the expression.

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

Step 12.2.1.1.7.1.7.5

Evaluate the exponent.

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3}{4})}{3}$

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

Step 12.2.1.1.7.1.8

Multiply $- 9$ by $3$ .

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 27}{4})}{3}$

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

$y = \frac{1 (\frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 27}{4})}{3}$

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

Step 12.2.1.1.7.2

Subtract $27$ from $9$ .

$y = \frac{1 (\frac{9 i \sqrt{3} + 9 i \sqrt{3} - 18}{4})}{3}$

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

Step 12.2.1.1.7.3

Add $9 i \sqrt{3}$ and $9 i \sqrt{3}$ .

$y = \frac{1 (\frac{18 i \sqrt{3} - 18}{4})}{3}$

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

$y = \frac{1 (\frac{18 i \sqrt{3} - 18}{4})}{3}$

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

Step 12.2.1.1.8

Reorder $18 i \sqrt{3}$ and $- 18$ .

$y = \frac{1 (\frac{- 18 + 18 i \sqrt{3}}{4})}{3}$

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

Step 12.2.1.1.9

Cancel the common factor of $- 18 + 18 i \sqrt{3}$ and $4$ .

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Step 12.2.1.1.9.1

Factor $2$ out of $- 18$ .

$y = \frac{1 (\frac{2 (- 9) + 18 i \sqrt{3}}{4})}{3}$

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

Step 12.2.1.1.9.2

Factor $2$ out of $18 i \sqrt{3}$ .

$y = \frac{1 (\frac{2 (- 9) + 2 (9 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.9.3

Factor $2$ out of $2 (- 9) + 2 (9 i \sqrt{3})$ .

$y = \frac{1 (\frac{2 (- 9 + 9 i \sqrt{3})}{4})}{3}$

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

Step 12.2.1.1.9.4

Cancel the common factors.

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Step 12.2.1.1.9.4.1

Factor $2$ out of $4$ .

$y = \frac{1 (\frac{2 (- 9 + 9 i \sqrt{3})}{2 \cdot 2})}{3}$

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

Step 12.2.1.1.9.4.2

Cancel the common factor.

$y = \frac{1 (\frac{2 (- 9 + 9 i \sqrt{3})}{2 \cdot 2})}{3}$

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

Step 12.2.1.1.9.4.3

Rewrite the expression.

$y = \frac{1 (\frac{- 9 + 9 i \sqrt{3}}{2})}{3}$

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

$y = \frac{1 (\frac{- 9 + 9 i \sqrt{3}}{2})}{3}$

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

$y = \frac{1 (\frac{- 9 + 9 i \sqrt{3}}{2})}{3}$

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

Step 12.2.1.1.10

Multiply $\frac{- 9 + 9 i \sqrt{3}}{2}$ by $1$ .

$y = \frac{\frac{- 9 + 9 i \sqrt{3}}{2}}{3}$

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

$y = \frac{\frac{- 9 + 9 i \sqrt{3}}{2}}{3}$

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

Step 12.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{- 9 + 9 i \sqrt{3}}{2} \cdot \frac{1}{3}$

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

Step 12.2.1.3

Multiply $\frac{- 9 + 9 i \sqrt{3}}{2} \cdot \frac{1}{3}$ .

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Step 12.2.1.3.1

Multiply $\frac{- 9 + 9 i \sqrt{3}}{2}$ by $\frac{1}{3}$ .

$y = \frac{- 9 + 9 i \sqrt{3}}{2 \cdot 3}$

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

Step 12.2.1.3.2

Multiply $2$ by $3$ .

$y = \frac{- 9 + 9 i \sqrt{3}}{6}$

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

$y = \frac{- 9 + 9 i \sqrt{3}}{6}$

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

Step 12.2.1.4

Cancel the common factor of $- 9 + 9 i \sqrt{3}$ and $6$ .

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Step 12.2.1.4.1

Factor $3$ out of $- 9$ .

$y = \frac{3 (- 3) + 9 i \sqrt{3}}{6}$

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

Step 12.2.1.4.2

Factor $3$ out of $9 i \sqrt{3}$ .

$y = \frac{3 (- 3) + 3 (3 i \sqrt{3})}{6}$

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

Step 12.2.1.4.3

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

$y = \frac{3 (- 3 + 3 i \sqrt{3})}{6}$

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

Step 12.2.1.4.4

Cancel the common factors.

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Step 12.2.1.4.4.1

Factor $3$ out of $6$ .

$y = \frac{3 (- 3 + 3 i \sqrt{3})}{3 \cdot 2}$

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

Step 12.2.1.4.4.2

Cancel the common factor.

$y = \frac{3 (- 3 + 3 i \sqrt{3})}{3 \cdot 2}$

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

Step 12.2.1.4.4.3

Rewrite the expression.

$y = \frac{- 3 + 3 i \sqrt{3}}{2}$

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

$y = \frac{- 3 + 3 i \sqrt{3}}{2}$

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

$y = \frac{- 3 + 3 i \sqrt{3}}{2}$

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

Step 12.2.1.5

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

$y = \frac{- 1 \cdot 3 + 3 i \sqrt{3}}{2}$

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

Step 12.2.1.6

Factor $- 1$ out of $3 i \sqrt{3}$ .

$y = \frac{- 1 \cdot 3 - (- 3 i \sqrt{3})}{2}$

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

Step 12.2.1.7

Factor $- 1$ out of $- 1 (3) - (- 3 i \sqrt{3})$ .

$y = \frac{- 1 (3 - 3 i \sqrt{3})}{2}$

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

Step 12.2.1.8

Move the negative in front of the fraction.

$y = - \frac{3 - 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 - 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 - 3 i \sqrt{3}}{2}$

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

$y = - \frac{3 - 3 i \sqrt{3}}{2}$

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

Step 13

List all of the solutions.

$y = 0, x = 0$

$y = 3, x = 3$

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

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

Step 14