Precalculus Examples

$4 x^{2} + 9 y^{2} = 72$ , $4 x - 3 y^{2} = 0$

Step 1

Solve for $x$ in $4 x - 3 y^{2} = 0$ .

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Step 1.1

Add $3 y^{2}$ to both sides of the equation.

$4 x = 3 y^{2}$

$4 x^{2} + 9 y^{2} = 72$

Step 1.2

Divide each term in $4 x = 3 y^{2}$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 3 y^{2}$ by $4$ .

$\frac{4 x}{4} = \frac{3 y^{2}}{4}$

$4 x^{2} + 9 y^{2} = 72$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3 y^{2}}{4}$

$4 x^{2} + 9 y^{2} = 72$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 y^{2}}{4}$

$4 x^{2} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$4 x^{2} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$4 x^{2} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$4 x^{2} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$4 x^{2} + 9 y^{2} = 72$

Step 2

Replace all occurrences of $x$ with $\frac{3 y^{2}}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x^{2} + 9 y^{2} = 72$ with $\frac{3 y^{2}}{4}$ .

$4 {(\frac{3 y^{2}}{4})}^{2} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1.1.1

Apply the product rule to $\frac{3 y^{2}}{4}$ .

$4 (\frac{{(3 y^{2})}^{2}}{4^{2}}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 2.2.1.1.2

Apply the product rule to $3 y^{2}$ .

$4 (\frac{3^{2} {(y^{2})}^{2}}{4^{2}}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$4 (\frac{3^{2} {(y^{2})}^{2}}{4^{2}}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 2.2.1.2

Simplify the numerator.

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Step 2.2.1.2.1

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

$4 (\frac{9 {(y^{2})}^{2}}{4^{2}}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 2.2.1.2.2

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

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Step 2.2.1.2.2.1

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

$4 (\frac{9 y^{2 \cdot 2}}{4^{2}}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 2.2.1.2.2.2

Multiply $2$ by $2$ .

$4 (\frac{9 y^{4}}{4^{2}}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$4 (\frac{9 y^{4}}{4^{2}}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$4 (\frac{9 y^{4}}{4^{2}}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 2.2.1.3

Raise $4$ to the power of $2$ .

$4 (\frac{9 y^{4}}{16}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 2.2.1.4

Cancel the common factor of $4$ .

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Step 2.2.1.4.1

Factor $4$ out of $16$ .

$4 (\frac{9 y^{4}}{4 (4)}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 2.2.1.4.2

Cancel the common factor.

$4 (\frac{9 y^{4}}{4 \cdot 4}) + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 2.2.1.4.3

Rewrite the expression.

$\frac{9 y^{4}}{4} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$\frac{9 y^{4}}{4} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$\frac{9 y^{4}}{4} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$\frac{9 y^{4}}{4} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

$\frac{9 y^{4}}{4} + 9 y^{2} = 72$

$x = \frac{3 y^{2}}{4}$

Step 3

Solve for $y$ in $\frac{9 y^{4}}{4} + 9 y^{2} = 72$ .

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Step 3.1

Multiply each term in $\frac{9 y^{4}}{4} + 9 y^{2} = 72$ by $4$ to eliminate the fractions.

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Step 3.1.1

Multiply each term in $\frac{9 y^{4}}{4} + 9 y^{2} = 72$ by $4$ .

$\frac{9 y^{4}}{4} \cdot 4 + 9 y^{2} \cdot 4 = 72 \cdot 4$

$x = \frac{3 y^{2}}{4}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Cancel the common factor of $4$ .

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Step 3.1.2.1.1.1

Cancel the common factor.

$\frac{9 y^{4}}{4} \cdot 4 + 9 y^{2} \cdot 4 = 72 \cdot 4$

$x = \frac{3 y^{2}}{4}$

Step 3.1.2.1.1.2

Rewrite the expression.

$9 y^{4} + 9 y^{2} \cdot 4 = 72 \cdot 4$

$x = \frac{3 y^{2}}{4}$

$9 y^{4} + 9 y^{2} \cdot 4 = 72 \cdot 4$

$x = \frac{3 y^{2}}{4}$

Step 3.1.2.1.2

Multiply $4$ by $9$ .

$9 y^{4} + 36 y^{2} = 72 \cdot 4$

$x = \frac{3 y^{2}}{4}$

$9 y^{4} + 36 y^{2} = 72 \cdot 4$

$x = \frac{3 y^{2}}{4}$

$9 y^{4} + 36 y^{2} = 72 \cdot 4$

$x = \frac{3 y^{2}}{4}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Multiply $72$ by $4$ .

$9 y^{4} + 36 y^{2} = 288$

$x = \frac{3 y^{2}}{4}$

$9 y^{4} + 36 y^{2} = 288$

$x = \frac{3 y^{2}}{4}$

$9 y^{4} + 36 y^{2} = 288$

$x = \frac{3 y^{2}}{4}$

Step 3.2

Subtract $288$ from both sides of the equation.

$9 y^{4} + 36 y^{2} - 288 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $9$ out of $9 y^{4} + 36 y^{2} - 288$ .

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Step 3.3.1.1

Factor $9$ out of $9 y^{4}$ .

$9 (y^{4}) + 36 y^{2} - 288 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.1.2

Factor $9$ out of $36 y^{2}$ .

$9 (y^{4}) + 9 (4 y^{2}) - 288 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.1.3

Factor $9$ out of $- 288$ .

$9 y^{4} + 9 (4 y^{2}) + 9 \cdot - 32 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.1.4

Factor $9$ out of $9 y^{4} + 9 (4 y^{2})$ .

$9 (y^{4} + 4 y^{2}) + 9 \cdot - 32 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.1.5

Factor $9$ out of $9 (y^{4} + 4 y^{2}) + 9 \cdot - 32$ .

$9 (y^{4} + 4 y^{2} - 32) = 0$

$x = \frac{3 y^{2}}{4}$

$9 (y^{4} + 4 y^{2} - 32) = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.2

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

$9 ({(y^{2})}^{2} + 4 y^{2} - 32) = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.3

Let $u = y^{2}$ . Substitute $u$ for all occurrences of $y^{2}$ .

$9 (u^{2} + 4 u - 32) = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.4

Factor $u^{2} + 4 u - 32$ using the AC method.

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Step 3.3.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 32$ and whose sum is $4$ .

$- 4, 8$

$x = \frac{3 y^{2}}{4}$

Step 3.3.4.2

Write the factored form using these integers.

$9 ((u - 4) (u + 8)) = 0$

$x = \frac{3 y^{2}}{4}$

$9 ((u - 4) (u + 8)) = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.5

Replace all occurrences of $u$ with $y^{2}$ .

$9 ((y^{2} - 4) (y^{2} + 8)) = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.6

Rewrite $4$ as $2^{2}$ .

$9 ((y^{2} - 2^{2}) (y^{2} + 8)) = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.7

Factor.

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Step 3.3.7.1

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

$9 ((y + 2) (y - 2) (y^{2} + 8)) = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.3.7.2

Remove unnecessary parentheses.

$9 (y + 2) (y - 2) (y^{2} + 8) = 0$

$x = \frac{3 y^{2}}{4}$

$9 (y + 2) (y - 2) (y^{2} + 8) = 0$

$x = \frac{3 y^{2}}{4}$

$9 (y + 2) (y - 2) (y^{2} + 8) = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.4

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

$y + 2 = 0$

$y - 2 = 0$

$y^{2} + 8 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.5

Set $y + 2$ equal to $0$ and solve for $y$ .

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Step 3.5.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.5.2

Subtract $2$ from both sides of the equation.

$y = - 2$

$x = \frac{3 y^{2}}{4}$

$y = - 2$

$x = \frac{3 y^{2}}{4}$

Step 3.6

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 3.6.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.6.2

Add $2$ to both sides of the equation.

$y = 2$

$x = \frac{3 y^{2}}{4}$

$y = 2$

$x = \frac{3 y^{2}}{4}$

Step 3.7

Set $y^{2} + 8$ equal to $0$ and solve for $y$ .

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Step 3.7.1

Set $y^{2} + 8$ equal to $0$ .

$y^{2} + 8 = 0$

$x = \frac{3 y^{2}}{4}$

Step 3.7.2

Solve $y^{2} + 8 = 0$ for $y$ .

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Step 3.7.2.1

Subtract $8$ from both sides of the equation.

$y^{2} = - 8$

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.2

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

$y = \pm \sqrt{- 8}$

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.3

Simplify $\pm \sqrt{- 8}$ .

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Step 3.7.2.3.1

Rewrite $- 8$ as $- 1 (8)$ .

$y = \pm \sqrt{- 1 \cdot 8}$

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.3.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{8}$

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.3.3

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

$y = \pm i \cdot \sqrt{8}$

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.3.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.7.2.3.4.1

Factor $4$ out of $8$ .

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

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.3.4.2

Rewrite $4$ as $2^{2}$ .

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

$x = \frac{3 y^{2}}{4}$

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

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.3.5

Pull terms out from under the radical.

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

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.3.6

Move $2$ to the left of $i$ .

$y = \pm 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

$y = \pm 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.4

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

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Step 3.7.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

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

$x = \frac{3 y^{2}}{4}$

Step 3.7.2.4.3

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

$y = 2 i \sqrt{2}, - 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

$y = 2 i \sqrt{2}, - 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

$y = 2 i \sqrt{2}, - 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

$y = 2 i \sqrt{2}, - 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

Step 3.8

The final solution is all the values that make $9 (y + 2) (y - 2) (y^{2} + 8) = 0$ true.

$y = - 2, 2, 2 i \sqrt{2}, - 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

$y = - 2, 2, 2 i \sqrt{2}, - 2 i \sqrt{2}$

$x = \frac{3 y^{2}}{4}$

Step 4

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $- 2$ .

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

$y = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3 {(- 2)}^{2}}{4}$ .

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Step 4.2.1.1

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

$x = \frac{3 \cdot 4}{4}$

$y = - 2$

Step 4.2.1.2

Multiply $3$ by $4$ .

$x = \frac{12}{4}$

$y = - 2$

Step 4.2.1.3

Divide $12$ by $4$ .

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

Step 5

Replace all occurrences of $y$ with $2$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $2$ .

$x = \frac{3 {(2)}^{2}}{4}$

$y = 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{3 {(2)}^{2}}{4}$ .

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Step 5.2.1.1

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

$x = \frac{3 \cdot 4}{4}$

$y = 2$

Step 5.2.1.2

Multiply $3$ by $4$ .

$x = \frac{12}{4}$

$y = 2$

Step 5.2.1.3

Divide $12$ by $4$ .

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

Step 6

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $- 2$ .

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

$y = - 2$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{3 {(- 2)}^{2}}{4}$ .

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Step 6.2.1.1

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

$x = \frac{3 \cdot 4}{4}$

$y = - 2$

Step 6.2.1.2

Multiply $3$ by $4$ .

$x = \frac{12}{4}$

$y = - 2$

Step 6.2.1.3

Divide $12$ by $4$ .

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

Step 7

Replace all occurrences of $y$ with $2$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $2$ .

$x = \frac{3 {(2)}^{2}}{4}$

$y = 2$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{3 {(2)}^{2}}{4}$ .

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Step 7.2.1.1

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

$x = \frac{3 \cdot 4}{4}$

$y = 2$

Step 7.2.1.2

Multiply $3$ by $4$ .

$x = \frac{12}{4}$

$y = 2$

Step 7.2.1.3

Divide $12$ by $4$ .

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

Step 8

Replace all occurrences of $y$ with $2 i \sqrt{2}$ in each equation.

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Step 8.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $2 i \sqrt{2}$ .

$x = \frac{3 {(2 i \sqrt{2})}^{2}}{4}$

$y = 2 i \sqrt{2}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{3 {(2 i \sqrt{2})}^{2}}{4}$ .

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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 $2 i \sqrt{2}$ .

$x = \frac{3 {(2 i)}^{2} {\sqrt{2}}^{2}}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.2

Apply the product rule to $2 i$ .

$x = \frac{3 (2^{2} i^{2}) {\sqrt{2}}^{2}}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.3

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

$x = \frac{3 (4 i^{2}) {\sqrt{2}}^{2}}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.4

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

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

$y = 2 i \sqrt{2}$

Step 8.2.1.1.5

Multiply $4$ by $- 1$ .

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

$y = 2 i \sqrt{2}$

Step 8.2.1.1.6

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

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Step 8.2.1.1.6.1

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

$x = \frac{3 \cdot (- 4 {(2^{\frac{1}{2}})}^{2})}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.6.2

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

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.6.3

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

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{2}{2}}}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.6.4

Cancel the common factor of $2$ .

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Step 8.2.1.1.6.4.1

Cancel the common factor.

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{2}{2}}}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.6.4.2

Rewrite the expression.

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.6.5

Evaluate the exponent.

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.7

Combine exponents.

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Step 8.2.1.1.7.1

Multiply $3$ by $- 4$ .

$x = \frac{- 12 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.1.7.2

Multiply $- 12$ by $2$ .

$x = \frac{- 24}{4}$

$y = 2 i \sqrt{2}$

$x = \frac{- 24}{4}$

$y = 2 i \sqrt{2}$

$x = \frac{- 24}{4}$

$y = 2 i \sqrt{2}$

Step 8.2.1.2

Divide $- 24$ by $4$ .

$x = - 6$

$y = 2 i \sqrt{2}$

$x = - 6$

$y = 2 i \sqrt{2}$

$x = - 6$

$y = 2 i \sqrt{2}$

$x = - 6$

$y = 2 i \sqrt{2}$

Step 9

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 9.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $- 2$ .

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

$y = - 2$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $\frac{3 {(- 2)}^{2}}{4}$ .

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Step 9.2.1.1

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

$x = \frac{3 \cdot 4}{4}$

$y = - 2$

Step 9.2.1.2

Multiply $3$ by $4$ .

$x = \frac{12}{4}$

$y = - 2$

Step 9.2.1.3

Divide $12$ by $4$ .

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

Step 10

Replace all occurrences of $y$ with $2$ in each equation.

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Step 10.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $2$ .

$x = \frac{3 {(2)}^{2}}{4}$

$y = 2$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $\frac{3 {(2)}^{2}}{4}$ .

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Step 10.2.1.1

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

$x = \frac{3 \cdot 4}{4}$

$y = 2$

Step 10.2.1.2

Multiply $3$ by $4$ .

$x = \frac{12}{4}$

$y = 2$

Step 10.2.1.3

Divide $12$ by $4$ .

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

Step 11

Replace all occurrences of $y$ with $2 i \sqrt{2}$ in each equation.

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Step 11.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $2 i \sqrt{2}$ .

$x = \frac{3 {(2 i \sqrt{2})}^{2}}{4}$

$y = 2 i \sqrt{2}$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $\frac{3 {(2 i \sqrt{2})}^{2}}{4}$ .

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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 $2 i \sqrt{2}$ .

$x = \frac{3 {(2 i)}^{2} {\sqrt{2}}^{2}}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.2

Apply the product rule to $2 i$ .

$x = \frac{3 (2^{2} i^{2}) {\sqrt{2}}^{2}}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.3

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

$x = \frac{3 (4 i^{2}) {\sqrt{2}}^{2}}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.4

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

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

$y = 2 i \sqrt{2}$

Step 11.2.1.1.5

Multiply $4$ by $- 1$ .

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

$y = 2 i \sqrt{2}$

Step 11.2.1.1.6

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

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Step 11.2.1.1.6.1

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

$x = \frac{3 \cdot (- 4 {(2^{\frac{1}{2}})}^{2})}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.6.2

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

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.6.3

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

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{2}{2}}}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.6.4

Cancel the common factor of $2$ .

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Step 11.2.1.1.6.4.1

Cancel the common factor.

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{2}{2}}}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.6.4.2

Rewrite the expression.

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.6.5

Evaluate the exponent.

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.7

Combine exponents.

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Step 11.2.1.1.7.1

Multiply $3$ by $- 4$ .

$x = \frac{- 12 \cdot 2}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.1.7.2

Multiply $- 12$ by $2$ .

$x = \frac{- 24}{4}$

$y = 2 i \sqrt{2}$

$x = \frac{- 24}{4}$

$y = 2 i \sqrt{2}$

$x = \frac{- 24}{4}$

$y = 2 i \sqrt{2}$

Step 11.2.1.2

Divide $- 24$ by $4$ .

$x = - 6$

$y = 2 i \sqrt{2}$

$x = - 6$

$y = 2 i \sqrt{2}$

$x = - 6$

$y = 2 i \sqrt{2}$

$x = - 6$

$y = 2 i \sqrt{2}$

Step 12

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

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Step 12.1

Replace all occurrences of $y$ in $x = \frac{3 y^{2}}{4}$ with $- 2 i \sqrt{2}$ .

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

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

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $\frac{3 {(- 2 i \sqrt{2})}^{2}}{4}$ .

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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 $- 2 i \sqrt{2}$ .

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

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

Step 12.2.1.1.2

Apply the product rule to $- 2 i$ .

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

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

Step 12.2.1.1.3

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

$x = \frac{3 (4 i^{2}) {\sqrt{2}}^{2}}{4}$

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

Step 12.2.1.1.4

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

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

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

Step 12.2.1.1.5

Multiply $4$ by $- 1$ .

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

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

Step 12.2.1.1.6

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

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Step 12.2.1.1.6.1

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

$x = \frac{3 \cdot (- 4 {(2^{\frac{1}{2}})}^{2})}{4}$

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

Step 12.2.1.1.6.2

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

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

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

Step 12.2.1.1.6.3

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

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{2}{2}}}{4}$

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

Step 12.2.1.1.6.4

Cancel the common factor of $2$ .

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Step 12.2.1.1.6.4.1

Cancel the common factor.

$x = \frac{3 \cdot - 4 \cdot 2^{\frac{2}{2}}}{4}$

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

Step 12.2.1.1.6.4.2

Rewrite the expression.

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

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

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

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

Step 12.2.1.1.6.5

Evaluate the exponent.

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

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

$x = \frac{3 \cdot - 4 \cdot 2}{4}$

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

Step 12.2.1.1.7

Combine exponents.

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Step 12.2.1.1.7.1

Multiply $3$ by $- 4$ .

$x = \frac{- 12 \cdot 2}{4}$

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

Step 12.2.1.1.7.2

Multiply $- 12$ by $2$ .

$x = \frac{- 24}{4}$

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

$x = \frac{- 24}{4}$

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

$x = \frac{- 24}{4}$

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

Step 12.2.1.2

Divide $- 24$ by $4$ .

$x = - 6$

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

$x = - 6$

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

$x = - 6$

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

$x = - 6$

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

Step 13

List all of the solutions.

$x = 3, y = - 2$

$x = 3, y = 2$

$x = - 6, y = 2 i \sqrt{2}$

$x = - 6, y = - 2 i \sqrt{2}$

Step 14