Precalculus Examples

2 x^{2} - 8 y^{3} = 19

$2 x^{2} - 8 y^{3} = 19$ ,

4 x^{2} + 16 y^{3} = 34

$4 x^{2} + 16 y^{3} = 34$

Step 1

Simplify the left side.

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Step 1.1

Reorder

2 x^{2}

$2 x^{2}$ and

- 8 y^{3}

$- 8 y^{3}$ .

- 8 y^{3} + 2 x^{2} = 19

$- 8 y^{3} + 2 x^{2} = 19$

4 x^{2} + 16 y^{3} = 34

$4 x^{2} + 16 y^{3} = 34$

- 8 y^{3} + 2 x^{2} = 19

$- 8 y^{3} + 2 x^{2} = 19$

4 x^{2} + 16 y^{3} = 34

$4 x^{2} + 16 y^{3} = 34$

Step 2

Simplify the left side.

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Step 2.1

Reorder

4 x^{2}

$4 x^{2}$ and

16 y^{3}

$16 y^{3}$ .

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

- 8 y^{3} + 2 x^{2} = 19

$- 8 y^{3} + 2 x^{2} = 19$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

- 8 y^{3} + 2 x^{2} = 19

$- 8 y^{3} + 2 x^{2} = 19$

Step 3

Multiply each equation by the value that makes the coefficients of

x^{2}

$x^{2}$ opposite.

- 2 \cdot (- 8 y^{3} + 2 x^{2}) = (- 2) (19)

$- 2 \cdot (- 8 y^{3} + 2 x^{2}) = (- 2) (19)$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify

- 2 \cdot (- 8 y^{3} + 2 x^{2})

$- 2 \cdot (- 8 y^{3} + 2 x^{2})$ .

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Step 4.1.1.1

Apply the distributive property.

- 2 (- 8 y^{3}) - 2 (2 x^{2}) = (- 2) (19)

$- 2 (- 8 y^{3}) - 2 (2 x^{2}) = (- 2) (19)$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply

- 8

$- 8$ by

- 2

$- 2$ .

16 y^{3} - 2 (2 x^{2}) = (- 2) (19)

$16 y^{3} - 2 (2 x^{2}) = (- 2) (19)$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

Step 4.1.1.2.2

Multiply

2

$2$ by

- 2

$- 2$ .

16 y^{3} - 4 x^{2} = (- 2) (19)

$16 y^{3} - 4 x^{2} = (- 2) (19)$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

16 y^{3} - 4 x^{2} = (- 2) (19)

$16 y^{3} - 4 x^{2} = (- 2) (19)$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

16 y^{3} - 4 x^{2} = (- 2) (19)

$16 y^{3} - 4 x^{2} = (- 2) (19)$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

16 y^{3} - 4 x^{2} = (- 2) (19)

$16 y^{3} - 4 x^{2} = (- 2) (19)$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply

- 2

$- 2$ by

19

$19$ .

16 y^{3} - 4 x^{2} = - 38

$16 y^{3} - 4 x^{2} = - 38$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

16 y^{3} - 4 x^{2} = - 38

$16 y^{3} - 4 x^{2} = - 38$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

16 y^{3} - 4 x^{2} = - 38

$16 y^{3} - 4 x^{2} = - 38$

16 y^{3} + 4 x^{2} = 34

$16 y^{3} + 4 x^{2} = 34$

Step 5

Add the two equations together to eliminate

x^{2}

$x^{2}$ from the system.

	$1$ $1$	$6$ $6$	$y^{3}$ $y^{3}$	$-$ $-$	$4$ $4$	$x^{2}$ $x^{2}$	$=$ $=$	$-$ $-$	$3$ $3$	$8$ $8$
$+$ $+$	$1$ $1$	$6$ $6$	$y^{3}$ $y^{3}$	$+$ $+$	$4$ $4$	$x^{2}$ $x^{2}$	$=$ $=$		$3$ $3$	$4$ $4$
	$3$ $3$	$2$ $2$	$y^{3}$ $y^{3}$				$=$ $=$		$-$ $-$	$4$ $4$

Step 6

Divide each term in

32 y^{3} = - 4

$32 y^{3} = - 4$ by

32

$32$ and simplify.

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Step 6.1

Divide each term in

32 y^{3} = - 4

$32 y^{3} = - 4$ by

32

$32$ .

\frac{32 y^{3}}{32} = \frac{- 4}{32}

$\frac{32 y^{3}}{32} = \frac{- 4}{32}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of

32

$32$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{32 y^{3}}{32} = \frac{- 4}{32}$

Step 6.2.1.2

Divide

$y^{3}$ by

$1$ .

$y^{3} = \frac{- 4}{32}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of

$- 4$ and

$32$ .

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Step 6.3.1.1

Factor

$4$ out of

$- 4$ .

$y^{3} = \frac{4 (- 1)}{32}$

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Factor

$4$ out of

$32$ .

$y^{3} = \frac{4 \cdot - 1}{4 \cdot 8}$

Step 6.3.1.2.2

Cancel the common factor.

$y^{3} = \frac{4 \cdot - 1}{4 \cdot 8}$

Step 6.3.1.2.3

Rewrite the expression.

$y^{3} = \frac{- 1}{8}$

Step 6.3.2

Move the negative in front of the fraction.

$y^{3} = - \frac{1}{8}$

Step 7

Substitute the value found for

$y^{3}$ into one of the original equations, then solve for

$x^{2}$ .

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Step 7.1

Substitute the value found for

$y^{3}$ into one of the original equations to solve for

$x^{2}$ .

$16 (- \frac{1}{8}) - 4 x^{2} = - 38$

Step 7.2

Simplify each term.

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Step 7.2.1

Cancel the common factor of

$8$ .

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Step 7.2.1.1

Move the leading negative in

$- \frac{1}{8}$ into the numerator.

$16 (\frac{- 1}{8}) - 4 x^{2} = - 38$

Step 7.2.1.2

Factor

$8$ out of

$16$ .

$8 (2) \frac{- 1}{8} - 4 x^{2} = - 38$

Step 7.2.1.3

Cancel the common factor.

$8 \cdot 2 \frac{- 1}{8} - 4 x^{2} = - 38$

Step 7.2.1.4

Rewrite the expression.

$2 \cdot - 1 - 4 x^{2} = - 38$

Step 7.2.2

Multiply

$2$ by

$- 1$ .

$- 2 - 4 x^{2} = - 38$

Step 7.3

Move all terms not containing

$x^{2}$ to the right side of the equation.

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Step 7.3.1

Add

$2$ to both sides of the equation.

$- 4 x^{2} = - 38 + 2$

Step 7.3.2

Add

$- 38$ and

$2$ .

$- 4 x^{2} = - 36$

Step 7.4

Divide each term in

$- 4 x^{2} = - 36$ by

$- 4$ and simplify.

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Step 7.4.1

Divide each term in

$- 4 x^{2} = - 36$ by

$- 4$ .

$\frac{- 4 x^{2}}{- 4} = \frac{- 36}{- 4}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of

$- 4$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{- 4 x^{2}}{- 4} = \frac{- 36}{- 4}$

Step 7.4.2.1.2

Divide

$x^{2}$ by

$1$ .

$x^{2} = \frac{- 36}{- 4}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Divide

$- 36$ by

$- 4$ .

$x^{2} = 9$

Step 8

This is the final solution to the independent system of equations.

$y^{3} = - \frac{1}{8}$

$x^{2} = 9$

Step 9

Add

$\frac{1}{8}$ to both sides of the equation.

$y^{3} + \frac{1}{8} = 0$

$x^{2} = 9$

Step 10

Factor the left side of the equation.

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Step 10.1

Rewrite

$1$ as

$1^{3}$ .

$y^{3} + \frac{1^{3}}{8} = 0$

$x^{2} = 9$

Step 10.2

Rewrite

$8$ as

$2^{3}$ .

$y^{3} + \frac{1^{3}}{2^{3}} = 0$

$x^{2} = 9$

Step 10.3

Rewrite

$\frac{1^{3}}{2^{3}}$ as

${(\frac{1}{2})}^{3}$ .

$y^{3} + {(\frac{1}{2})}^{3} = 0$

$x^{2} = 9$

Step 10.4

Since both terms are perfect cubes, factor using the sum of cubes formula,

$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where

$a = y$ and

$b = \frac{1}{2}$ .

$(y + \frac{1}{2}) (y^{2} - y \frac{1}{2} + {(\frac{1}{2})}^{2}) = 0$

$x^{2} = 9$

Step 10.5

Simplify.

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Step 10.5.1

Combine

$\frac{1}{2}$ and

$y$ .

$(y + \frac{1}{2}) (y^{2} - \frac{y}{2} + {(\frac{1}{2})}^{2}) = 0$

$x^{2} = 9$

Step 10.5.2

Apply the product rule to

$\frac{1}{2}$ .

$(y + \frac{1}{2}) (y^{2} - \frac{y}{2} + \frac{1^{2}}{2^{2}}) = 0$

$x^{2} = 9$

Step 10.5.3

One to any power is one.

$(y + \frac{1}{2}) (y^{2} - \frac{y}{2} + \frac{1}{2^{2}}) = 0$

$x^{2} = 9$

Step 10.5.4

Raise

$2$ to the power of

$2$ .

$(y + \frac{1}{2}) (y^{2} - \frac{y}{2} + \frac{1}{4}) = 0$

$x^{2} = 9$

$(y + \frac{1}{2}) (y^{2} - \frac{y}{2} + \frac{1}{4}) = 0$

$x^{2} = 9$

$(y + \frac{1}{2}) (y^{2} - \frac{y}{2} + \frac{1}{4}) = 0$

$x^{2} = 9$

Step 11

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$y + \frac{1}{2} = 0$

$y^{2} - \frac{y}{2} + \frac{1}{4} = 0$

$x^{2} = 9$

Step 12

Set

$y + \frac{1}{2}$ equal to

$0$ and solve for

$y$ .

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Step 12.1

Set

$y + \frac{1}{2}$ equal to

$0$ .

$y + \frac{1}{2} = 0$

$x^{2} = 9$

Step 12.2

Subtract

$\frac{1}{2}$ from both sides of the equation.

$y = - \frac{1}{2}$

$x^{2} = 9$

$y = - \frac{1}{2}$

$x^{2} = 9$

Step 13

Set

$y^{2} - \frac{y}{2} + \frac{1}{4}$ equal to

$0$ and solve for

$y$ .

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Step 13.1

Set

$y^{2} - \frac{y}{2} + \frac{1}{4}$ equal to

$0$ .

$y^{2} - \frac{y}{2} + \frac{1}{4} = 0$

$x^{2} = 9$

Step 13.2

Solve

$y^{2} - \frac{y}{2} + \frac{1}{4} = 0$ for

$y$ .

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Step 13.2.1

Multiply through by the least common denominator

$4$ , then simplify.

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Step 13.2.1.1

Apply the distributive property.

$4 y^{2} + 4 (- \frac{y}{2}) + 4 (\frac{1}{4}) = 0$

$x^{2} = 9$

Step 13.2.1.2

Simplify.

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Step 13.2.1.2.1

Cancel the common factor of

$2$ .

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Step 13.2.1.2.1.1

Move the leading negative in

$- \frac{y}{2}$ into the numerator.

$4 y^{2} + 4 (\frac{- y}{2}) + 4 (\frac{1}{4}) = 0$

$x^{2} = 9$

Step 13.2.1.2.1.2

Factor

$2$ out of

$4$ .

$4 y^{2} + 2 (2) (\frac{- y}{2}) + 4 (\frac{1}{4}) = 0$

$x^{2} = 9$

Step 13.2.1.2.1.3

Cancel the common factor.

$4 y^{2} + 2 \cdot (2 (\frac{- y}{2})) + 4 (\frac{1}{4}) = 0$

$x^{2} = 9$

Step 13.2.1.2.1.4

Rewrite the expression.

$4 y^{2} + 2 (- y) + 4 (\frac{1}{4}) = 0$

$x^{2} = 9$

$4 y^{2} + 2 (- y) + 4 (\frac{1}{4}) = 0$

$x^{2} = 9$

Step 13.2.1.2.2

Multiply

$- 1$ by

$2$ .

$4 y^{2} - 2 y + 4 (\frac{1}{4}) = 0$

$x^{2} = 9$

Step 13.2.1.2.3

Cancel the common factor of

$4$ .

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Step 13.2.1.2.3.1

Cancel the common factor.

$4 y^{2} - 2 y + 4 (\frac{1}{4}) = 0$

$x^{2} = 9$

Step 13.2.1.2.3.2

Rewrite the expression.

$4 y^{2} - 2 y + 1 = 0$

$x^{2} = 9$

$4 y^{2} - 2 y + 1 = 0$

$x^{2} = 9$

$4 y^{2} - 2 y + 1 = 0$

$x^{2} = 9$

$4 y^{2} - 2 y + 1 = 0$

$x^{2} = 9$

Step 13.2.2

Use the quadratic formula to find the solutions.

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

$x^{2} = 9$

Step 13.2.3

Substitute the values

$a = 4$ ,

$b = - 2$ , and

$c = 1$ into the quadratic formula and solve for

$y$ .

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

$x^{2} = 9$

Step 13.2.4

Simplify.

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Step 13.2.4.1

Simplify the numerator.

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Step 13.2.4.1.1

Raise

$- 2$ to the power of

$2$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.2

Multiply

$- 4 \cdot 4 \cdot 1$ .

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Step 13.2.4.1.2.1

Multiply

$- 4$ by

$4$ .

$y = \frac{2 \pm \sqrt{4 - 16 \cdot 1}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.2.2

Multiply

$- 16$ by

$1$ .

$y = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.3

Subtract

$16$ from

$4$ .

$y = \frac{2 \pm \sqrt{- 12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.4

Rewrite

$- 12$ as

$- 1 (12)$ .

$y = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.5

Rewrite

$\sqrt{- 1 (12)}$ as

$\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$y = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.7

Rewrite

$12$ as

$2^{2} \cdot 3$ .

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Step 13.2.4.1.7.1

Factor

$4$ out of

$12$ .

$y = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.7.2

Rewrite

$4$ as

$2^{2}$ .

$y = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.8

Pull terms out from under the radical.

$y = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.1.9

Move

$2$ to the left of

$i$ .

$y = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.4.2

Multiply

$2$ by

$4$ .

$y = \frac{2 \pm 2 i \sqrt{3}}{8}$

$x^{2} = 9$

Step 13.2.4.3

Simplify

$\frac{2 \pm 2 i \sqrt{3}}{8}$ .

$y = \frac{1 \pm i \sqrt{3}}{4}$

$x^{2} = 9$

$y = \frac{1 \pm i \sqrt{3}}{4}$

$x^{2} = 9$

Step 13.2.5

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 13.2.5.1

Simplify the numerator.

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Step 13.2.5.1.1

Raise

$- 2$ to the power of

$2$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.2

Multiply

$- 4 \cdot 4 \cdot 1$ .

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Step 13.2.5.1.2.1

Multiply

$- 4$ by

$4$ .

$y = \frac{2 \pm \sqrt{4 - 16 \cdot 1}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.2.2

Multiply

$- 16$ by

$1$ .

$y = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.3

Subtract

$16$ from

$4$ .

$y = \frac{2 \pm \sqrt{- 12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.4

Rewrite

$- 12$ as

$- 1 (12)$ .

$y = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.5

Rewrite

$\sqrt{- 1 (12)}$ as

$\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$y = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.7

Rewrite

$12$ as

$2^{2} \cdot 3$ .

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Step 13.2.5.1.7.1

Factor

$4$ out of

$12$ .

$y = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.7.2

Rewrite

$4$ as

$2^{2}$ .

$y = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.8

Pull terms out from under the radical.

$y = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.1.9

Move

$2$ to the left of

$i$ .

$y = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.5.2

Multiply

$2$ by

$4$ .

$y = \frac{2 \pm 2 i \sqrt{3}}{8}$

$x^{2} = 9$

Step 13.2.5.3

Simplify

$\frac{2 \pm 2 i \sqrt{3}}{8}$ .

$y = \frac{1 \pm i \sqrt{3}}{4}$

$x^{2} = 9$

Step 13.2.5.4

Change the

$\pm$ to

$+$ .

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

$x^{2} = 9$

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

$x^{2} = 9$

Step 13.2.6

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 13.2.6.1

Simplify the numerator.

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Step 13.2.6.1.1

Raise

$- 2$ to the power of

$2$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.2

Multiply

$- 4 \cdot 4 \cdot 1$ .

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Step 13.2.6.1.2.1

Multiply

$- 4$ by

$4$ .

$y = \frac{2 \pm \sqrt{4 - 16 \cdot 1}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.2.2

Multiply

$- 16$ by

$1$ .

$y = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.3

Subtract

$16$ from

$4$ .

$y = \frac{2 \pm \sqrt{- 12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.4

Rewrite

$- 12$ as

$- 1 (12)$ .

$y = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.5

Rewrite

$\sqrt{- 1 (12)}$ as

$\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$y = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.7

Rewrite

$12$ as

$2^{2} \cdot 3$ .

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Step 13.2.6.1.7.1

Factor

$4$ out of

$12$ .

$y = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.7.2

Rewrite

$4$ as

$2^{2}$ .

$y = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.8

Pull terms out from under the radical.

$y = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.1.9

Move

$2$ to the left of

$i$ .

$y = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

$x^{2} = 9$

$y = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 4}$

$x^{2} = 9$

Step 13.2.6.2

Multiply

$2$ by

$4$ .

$y = \frac{2 \pm 2 i \sqrt{3}}{8}$

$x^{2} = 9$

Step 13.2.6.3

Simplify

$\frac{2 \pm 2 i \sqrt{3}}{8}$ .

$y = \frac{1 \pm i \sqrt{3}}{4}$

$x^{2} = 9$

Step 13.2.6.4

Change the

$\pm$ to

$-$ .

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

$x^{2} = 9$

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

$x^{2} = 9$

Step 13.2.7

The final answer is the combination of both solutions.

$y = \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

$x^{2} = 9$

$y = \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

$x^{2} = 9$

$y = \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

$x^{2} = 9$

Step 14

The final solution is all the values that make

$(y + \frac{1}{2}) (y^{2} - \frac{y}{2} + \frac{1}{4}) = 0$ true.

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

$x^{2} = 9$

Step 15

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

Step 16

Simplify

$\pm \sqrt{9}$ .

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Step 16.1

Rewrite

$9$ as

$3^{2}$ .

$x = \pm \sqrt{3^{2}}$

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

Step 16.2

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

$x = \pm 3$

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

$x = \pm 3$

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

Step 17

The complete solution is the result of both the positive and negative portions of the solution.

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Step 17.1

First, use the positive value of the

$\pm$ to find the first solution.

$x = 3$

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

Step 17.2

Next, use the negative value of the

$\pm$ to find the second solution.

$x = - 3$

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

Step 17.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

$x = 3, - 3$

$y = - \frac{1}{2}, \frac{1 + i \sqrt{3}}{4}, \frac{1 - i \sqrt{3}}{4}$

Step 18

The final result is the combination of all values of

$x$ with all values of

$y$ .

$(3, - \frac{1}{2}), (3, \frac{1 + i \sqrt{3}}{4}), (3, \frac{1 - i \sqrt{3}}{4}), (- 3, - \frac{1}{2}), (- 3, \frac{1 + i \sqrt{3}}{4}), (- 3, \frac{1 - i \sqrt{3}}{4})$

Step 19