Precalculus Examples

$4 x^{2} + 3 y^{2} = 48$ , $3 x^{2} - 5 x^{2} = 7$

Step 1

Solve for $x$ in $- 2 x^{2} = 7$ .

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Step 1.1

Divide each term in $- 2 x^{2} = 7$ by $- 2$ and simplify.

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Step 1.1.1

Divide each term in $- 2 x^{2} = 7$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{7}{- 2}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $- 2$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{7}{- 2}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{7}{- 2}$

$4 x^{2} + 3 y^{2} = 48$

$x^{2} = \frac{7}{- 2}$

$4 x^{2} + 3 y^{2} = 48$

$x^{2} = \frac{7}{- 2}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{7}{2}$

$4 x^{2} + 3 y^{2} = 48$

$x^{2} = - \frac{7}{2}$

$4 x^{2} + 3 y^{2} = 48$

$x^{2} = - \frac{7}{2}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.2

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

$x = \pm \sqrt{- \frac{7}{2}}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3

Simplify $\pm \sqrt{- \frac{7}{2}}$ .

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Step 1.3.1

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

$x = \pm \sqrt{i^{2} (\frac{7}{2})}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{7}{2}}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.3

Rewrite $\sqrt{\frac{7}{2}}$ as $\frac{\sqrt{7}}{\sqrt{2}}$ .

$x = \pm i (\frac{\sqrt{7}}{\sqrt{2}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.4

Multiply $\frac{\sqrt{7}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{\sqrt{7}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5

Combine and simplify the denominator.

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Step 1.3.5.1

Multiply $\frac{\sqrt{7}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.4

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

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{{\sqrt{2}}^{1 + 1}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.5

Add $1$ and $1$ .

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{{\sqrt{2}}^{2}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.6

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

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Step 1.3.5.6.1

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

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.6.2

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

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.6.3

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

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{2^{\frac{2}{2}}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.6.4

Cancel the common factor of $2$ .

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Step 1.3.5.6.4.1

Cancel the common factor.

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{2^{\frac{2}{2}}})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.6.4.2

Rewrite the expression.

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{2})$

$4 x^{2} + 3 y^{2} = 48$

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{2})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.5.6.5

Evaluate the exponent.

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{2})$

$4 x^{2} + 3 y^{2} = 48$

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{2})$

$4 x^{2} + 3 y^{2} = 48$

$x = \pm i (\frac{\sqrt{7} \sqrt{2}}{2})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Combine using the product rule for radicals.

$x = \pm i (\frac{\sqrt{7 \cdot 2}}{2})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.6.2

Multiply $7$ by $2$ .

$x = \pm i (\frac{\sqrt{14}}{2})$

$4 x^{2} + 3 y^{2} = 48$

$x = \pm i (\frac{\sqrt{14}}{2})$

$4 x^{2} + 3 y^{2} = 48$

Step 1.3.7

Combine $i$ and $\frac{\sqrt{14}}{2}$ .

$x = \pm \frac{i \sqrt{14}}{2}$

$4 x^{2} + 3 y^{2} = 48$

$x = \pm \frac{i \sqrt{14}}{2}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.4

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

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Step 1.4.1

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

$x = \frac{i \sqrt{14}}{2}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.4.2

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

$x = - \frac{i \sqrt{14}}{2}$

$4 x^{2} + 3 y^{2} = 48$

Step 1.4.3

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

$x = \frac{i \sqrt{14}}{2}, - \frac{i \sqrt{14}}{2}$

$4 x^{2} + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}, - \frac{i \sqrt{14}}{2}$

$4 x^{2} + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}, - \frac{i \sqrt{14}}{2}$

$4 x^{2} + 3 y^{2} = 48$

Step 2

Solve the system $x = \frac{i \sqrt{14}}{2}, 4 x^{2} + 3 y^{2} = 48$ .

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Step 2.1

Replace all occurrences of $x$ with $\frac{i \sqrt{14}}{2}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $4 x^{2} + 3 y^{2} = 48$ with $\frac{i \sqrt{14}}{2}$ .

$4 {(\frac{i \sqrt{14}}{2})}^{2} + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

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

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Step 2.1.2.1.1.1

Apply the product rule to $\frac{i \sqrt{14}}{2}$ .

$4 (\frac{{(i \sqrt{14})}^{2}}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.1.2

Apply the product rule to $i \sqrt{14}$ .

$4 (\frac{i^{2} {\sqrt{14}}^{2}}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

$4 (\frac{i^{2} {\sqrt{14}}^{2}}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.2

Simplify the numerator.

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Step 2.1.2.1.2.1

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

$4 (\frac{- 1 {\sqrt{14}}^{2}}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.2.2

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

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Step 2.1.2.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$4 (\frac{- 1 {(14^{\frac{1}{2}})}^{2}}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.2.2.2

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

$4 (\frac{- 1 \cdot 14^{\frac{1}{2} \cdot 2}}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.2.2.3

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

$4 (\frac{- 1 \cdot 14^{\frac{2}{2}}}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.2.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.2.2.4.1

Cancel the common factor.

$4 (\frac{- 1 \cdot 14^{\frac{2}{2}}}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.2.2.4.2

Rewrite the expression.

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.2.2.5

Evaluate the exponent.

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.3

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

$4 (\frac{- 1 \cdot 14}{4}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.4

Multiply $- 1$ by $14$ .

$4 (\frac{- 14}{4}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.5

Cancel the common factor of $4$ .

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Step 2.1.2.1.5.1

Cancel the common factor.

$4 (\frac{- 14}{4}) + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.1.2.1.5.2

Rewrite the expression.

$- 14 + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

$- 14 + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

$- 14 + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

$- 14 + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

$- 14 + 3 y^{2} = 48$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2

Solve for $y$ in $- 14 + 3 y^{2} = 48$ .

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Step 2.2.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.1.1

Add $14$ to both sides of the equation.

$3 y^{2} = 48 + 14$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.1.2

Add $48$ and $14$ .

$3 y^{2} = 62$

$x = \frac{i \sqrt{14}}{2}$

$3 y^{2} = 62$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.2

Divide each term in $3 y^{2} = 62$ by $3$ and simplify.

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Step 2.2.2.1

Divide each term in $3 y^{2} = 62$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{62}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{62}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{62}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y^{2} = \frac{62}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y^{2} = \frac{62}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y^{2} = \frac{62}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.3

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

$y = \pm \sqrt{\frac{62}{3}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4

Simplify $\pm \sqrt{\frac{62}{3}}$ .

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Step 2.2.4.1

Rewrite $\sqrt{\frac{62}{3}}$ as $\frac{\sqrt{62}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{62}}{\sqrt{3}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.2

Multiply $\frac{\sqrt{62}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{62}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3

Combine and simplify the denominator.

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Step 2.2.4.3.1

Multiply $\frac{\sqrt{62}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{62} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.2

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.3

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.4

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{62} \sqrt{3}}{{\sqrt{3}}^{2}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.6

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

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Step 2.2.4.3.6.1

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.6.2

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.6.3

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 2.2.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.4

Simplify the numerator.

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Step 2.2.4.4.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{62 \cdot 3}}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.4.4.2

Multiply $62$ by $3$ .

$y = \pm \frac{\sqrt{186}}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{186}}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{186}}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.5

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

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Step 2.2.5.1

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

$y = \frac{\sqrt{186}}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.5.2

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

$y = - \frac{\sqrt{186}}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.2.5.3

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

$y = \frac{\sqrt{186}}{3}, - \frac{\sqrt{186}}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y = \frac{\sqrt{186}}{3}, - \frac{\sqrt{186}}{3}$

$x = \frac{i \sqrt{14}}{2}$

$y = \frac{\sqrt{186}}{3}, - \frac{\sqrt{186}}{3}$

$x = \frac{i \sqrt{14}}{2}$

Step 2.3

Solve the system of equations.

$y = \frac{\sqrt{186}}{3}, x = \frac{i \sqrt{14}}{2}$

Step 2.4

Solve the system of equations.

$y = - \frac{\sqrt{186}}{3}, x = \frac{i \sqrt{14}}{2}$

Step 3

Solve the system $x = - \frac{i \sqrt{14}}{2}, 4 x^{2} + 3 y^{2} = 48$ .

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Step 3.1

Replace all occurrences of $x$ with $- \frac{i \sqrt{14}}{2}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $4 x^{2} + 3 y^{2} = 48$ with $- \frac{i \sqrt{14}}{2}$ .

$4 {(- \frac{i \sqrt{14}}{2})}^{2} + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

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

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

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Step 3.1.2.1.1.1

Apply the product rule to $- \frac{i \sqrt{14}}{2}$ .

$4 ({(- 1)}^{2} {(\frac{i \sqrt{14}}{2})}^{2}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.1.2

Apply the product rule to $\frac{i \sqrt{14}}{2}$ .

$4 ({(- 1)}^{2} (\frac{{(i \sqrt{14})}^{2}}{2^{2}})) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.1.3

Apply the product rule to $i \sqrt{14}$ .

$4 ({(- 1)}^{2} (\frac{i^{2} {\sqrt{14}}^{2}}{2^{2}})) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

$4 ({(- 1)}^{2} (\frac{i^{2} {\sqrt{14}}^{2}}{2^{2}})) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.2

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

$4 (1 (\frac{i^{2} {\sqrt{14}}^{2}}{2^{2}})) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.3

Multiply $\frac{i^{2} {\sqrt{14}}^{2}}{2^{2}}$ by $1$ .

$4 (\frac{i^{2} {\sqrt{14}}^{2}}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.4

Simplify the numerator.

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Step 3.1.2.1.4.1

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

$4 (\frac{- 1 {\sqrt{14}}^{2}}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.4.2

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

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Step 3.1.2.1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$4 (\frac{- 1 {(14^{\frac{1}{2}})}^{2}}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.4.2.2

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

$4 (\frac{- 1 \cdot 14^{\frac{1}{2} \cdot 2}}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.4.2.3

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

$4 (\frac{- 1 \cdot 14^{\frac{2}{2}}}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.4.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.4.2.4.1

Cancel the common factor.

$4 (\frac{- 1 \cdot 14^{\frac{2}{2}}}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.4.2.4.2

Rewrite the expression.

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.4.2.5

Evaluate the exponent.

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

$4 (\frac{- 1 \cdot 14}{2^{2}}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.5

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

$4 (\frac{- 1 \cdot 14}{4}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.6

Multiply $- 1$ by $14$ .

$4 (\frac{- 14}{4}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.7

Cancel the common factor of $4$ .

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Step 3.1.2.1.7.1

Cancel the common factor.

$4 (\frac{- 14}{4}) + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.1.2.1.7.2

Rewrite the expression.

$- 14 + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

$- 14 + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

$- 14 + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

$- 14 + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

$- 14 + 3 y^{2} = 48$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2

Solve for $y$ in $- 14 + 3 y^{2} = 48$ .

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Step 3.2.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.2.1.1

Add $14$ to both sides of the equation.

$3 y^{2} = 48 + 14$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.1.2

Add $48$ and $14$ .

$3 y^{2} = 62$

$x = - \frac{i \sqrt{14}}{2}$

$3 y^{2} = 62$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.2

Divide each term in $3 y^{2} = 62$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 y^{2} = 62$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{62}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{62}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{62}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y^{2} = \frac{62}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y^{2} = \frac{62}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y^{2} = \frac{62}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.3

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

$y = \pm \sqrt{\frac{62}{3}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4

Simplify $\pm \sqrt{\frac{62}{3}}$ .

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Step 3.2.4.1

Rewrite $\sqrt{\frac{62}{3}}$ as $\frac{\sqrt{62}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{62}}{\sqrt{3}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.2

Multiply $\frac{\sqrt{62}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{62}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3

Combine and simplify the denominator.

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Step 3.2.4.3.1

Multiply $\frac{\sqrt{62}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{62} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.2

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.3

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.4

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{62} \sqrt{3}}{{\sqrt{3}}^{2}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.6

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

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Step 3.2.4.3.6.1

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.6.2

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.6.3

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

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3^{\frac{2}{2}}}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{62} \sqrt{3}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.4

Simplify the numerator.

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Step 3.2.4.4.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{62 \cdot 3}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.4.4.2

Multiply $62$ by $3$ .

$y = \pm \frac{\sqrt{186}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{186}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y = \pm \frac{\sqrt{186}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.5

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

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Step 3.2.5.1

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

$y = \frac{\sqrt{186}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.5.2

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

$y = - \frac{\sqrt{186}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.2.5.3

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

$y = \frac{\sqrt{186}}{3}, - \frac{\sqrt{186}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y = \frac{\sqrt{186}}{3}, - \frac{\sqrt{186}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

$y = \frac{\sqrt{186}}{3}, - \frac{\sqrt{186}}{3}$

$x = - \frac{i \sqrt{14}}{2}$

Step 3.3

Solve the system of equations.

$y = \frac{\sqrt{186}}{3}, x = - \frac{i \sqrt{14}}{2}$

Step 3.4

Solve the system of equations.

$y = - \frac{\sqrt{186}}{3}, x = - \frac{i \sqrt{14}}{2}$

Step 4

List all of the solutions.

$y = \frac{\sqrt{186}}{3}, x = \frac{i \sqrt{14}}{2}$

$y = - \frac{\sqrt{186}}{3}, x = \frac{i \sqrt{14}}{2}$

$y = \frac{\sqrt{186}}{3}, x = - \frac{i \sqrt{14}}{2}$

$y = - \frac{\sqrt{186}}{3}, x = - \frac{i \sqrt{14}}{2}$

Step 5