Precalculus Examples

$2 x^{2} - 7 y^{2} + 10 = 0$ , $x^{2} - 8 y^{2} + 5 = 0$

Step 1

Solve for $x$ in $x^{2} - 8 y^{2} + 5 = 0$ .

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Step 1.1

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

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Step 1.1.1

Add $8 y^{2}$ to both sides of the equation.

$x^{2} + 5 = 8 y^{2}$

$2 x^{2} - 7 y^{2} + 10 = 0$

Step 1.1.2

Subtract $5$ from both sides of the equation.

$x^{2} = 8 y^{2} - 5$

$2 x^{2} - 7 y^{2} + 10 = 0$

$x^{2} = 8 y^{2} - 5$

$2 x^{2} - 7 y^{2} + 10 = 0$

Step 1.2

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

$x = \pm \sqrt{8 y^{2} - 5}$

$2 x^{2} - 7 y^{2} + 10 = 0$

Step 1.3

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

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Step 1.3.1

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

$x = \sqrt{8 y^{2} - 5}$

$2 x^{2} - 7 y^{2} + 10 = 0$

Step 1.3.2

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

$x = - \sqrt{8 y^{2} - 5}$

$2 x^{2} - 7 y^{2} + 10 = 0$

Step 1.3.3

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

$x = \sqrt{8 y^{2} - 5}$

$x = - \sqrt{8 y^{2} - 5}$

$2 x^{2} - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

$x = - \sqrt{8 y^{2} - 5}$

$2 x^{2} - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

$x = - \sqrt{8 y^{2} - 5}$

$2 x^{2} - 7 y^{2} + 10 = 0$

Step 2

Solve the system $x = \sqrt{8 y^{2} - 5}, 2 x^{2} - 7 y^{2} + 10 = 0$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{8 y^{2} - 5}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $2 x^{2} - 7 y^{2} + 10 = 0$ with $\sqrt{8 y^{2} - 5}$ .

$2 {(\sqrt{8 y^{2} - 5})}^{2} - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $2 {(\sqrt{8 y^{2} - 5})}^{2} - 7 y^{2} + 10$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Rewrite ${\sqrt{8 y^{2} - 5}}^{2}$ as $8 y^{2} - 5$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 y^{2} - 5}$ as ${(8 y^{2} - 5)}^{\frac{1}{2}}$ .

$2 {({(8 y^{2} - 5)}^{\frac{1}{2}})}^{2} - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.1.1.2

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

$2 {(8 y^{2} - 5)}^{\frac{1}{2} \cdot 2} - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.1.1.3

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

$2 {(8 y^{2} - 5)}^{\frac{2}{2}} - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.1.4.1

Cancel the common factor.

$2 {(8 y^{2} - 5)}^{\frac{2}{2}} - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$2 (8 y^{2} - 5) - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

$2 (8 y^{2} - 5) - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.1.1.5

Simplify.

$2 (8 y^{2} - 5) - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

$2 (8 y^{2} - 5) - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.1.2

Apply the distributive property.

$2 (8 y^{2}) + 2 \cdot - 5 - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.1.3

Multiply $8$ by $2$ .

$16 y^{2} + 2 \cdot - 5 - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.1.4

Multiply $2$ by $- 5$ .

$16 y^{2} - 10 - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

$16 y^{2} - 10 - 7 y^{2} + 10 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.2

Simplify by adding terms.

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Step 2.1.2.1.2.1

Combine the opposite terms in $16 y^{2} - 10 - 7 y^{2} + 10$ .

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Step 2.1.2.1.2.1.1

Add $- 10$ and $10$ .

$16 y^{2} - 7 y^{2} + 0 = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.2.1.2

Add $16 y^{2} - 7 y^{2}$ and $0$ .

$16 y^{2} - 7 y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

$16 y^{2} - 7 y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.1.2.1.2.2

Subtract $7 y^{2}$ from $16 y^{2}$ .

$9 y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

$9 y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

$9 y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

$9 y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

$9 y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.2

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

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Step 2.2.1

Divide each term in $9 y^{2} = 0$ by $9$ and simplify.

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Step 2.2.1.1

Divide each term in $9 y^{2} = 0$ by $9$ .

$\frac{9 y^{2}}{9} = \frac{0}{9}$

$x = \sqrt{8 y^{2} - 5}$

Step 2.2.1.2

Simplify the left side.

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Step 2.2.1.2.1

Cancel the common factor of $9$ .

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Step 2.2.1.2.1.1

Cancel the common factor.

$\frac{9 y^{2}}{9} = \frac{0}{9}$

$x = \sqrt{8 y^{2} - 5}$

Step 2.2.1.2.1.2

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

$y^{2} = \frac{0}{9}$

$x = \sqrt{8 y^{2} - 5}$

$y^{2} = \frac{0}{9}$

$x = \sqrt{8 y^{2} - 5}$

$y^{2} = \frac{0}{9}$

$x = \sqrt{8 y^{2} - 5}$

Step 2.2.1.3

Simplify the right side.

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Step 2.2.1.3.1

Divide $0$ by $9$ .

$y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

$y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

$y^{2} = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.2.2

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

$y = \pm \sqrt{0}$

$x = \sqrt{8 y^{2} - 5}$

Step 2.2.3

Simplify $\pm \sqrt{0}$ .

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Step 2.2.3.1

Rewrite $0$ as $0^{2}$ .

$y = \pm \sqrt{0^{2}}$

$x = \sqrt{8 y^{2} - 5}$

Step 2.2.3.2

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

$y = \pm 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.2.3.3

Plus or minus $0$ is $0$ .

$y = 0$

$x = \sqrt{8 y^{2} - 5}$

$y = 0$

$x = \sqrt{8 y^{2} - 5}$

$y = 0$

$x = \sqrt{8 y^{2} - 5}$

Step 2.3

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

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{8 y^{2} - 5}$ with $0$ .

$x = \sqrt{8 {(0)}^{2} - 5}$

$y = 0$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{8 {(0)}^{2} - 5}$ .

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Step 2.3.2.1.1

Simplify the expression.

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Step 2.3.2.1.1.1

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

$x = \sqrt{8 \cdot 0 - 5}$

$y = 0$

Step 2.3.2.1.1.2

Multiply $8$ by $0$ .

$x = \sqrt{0 - 5}$

$y = 0$

Step 2.3.2.1.1.3

Subtract $5$ from $0$ .

$x = \sqrt{- 5}$

$y = 0$

$x = \sqrt{- 5}$

$y = 0$

Step 2.3.2.1.2

Rewrite $- 5$ as $- 1 (5)$ .

$x = \sqrt{- 1 \cdot 5}$

$y = 0$

Step 2.3.2.1.3

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$x = \sqrt{- 1} \cdot \sqrt{5}$

$y = 0$

Step 2.3.2.1.4

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

$x = i \sqrt{5}$

$y = 0$

$x = i \sqrt{5}$

$y = 0$

$x = i \sqrt{5}$

$y = 0$

$x = i \sqrt{5}$

$y = 0$

$x = i \sqrt{5}$

$y = 0$

Step 3

Solve the system $x = - \sqrt{8 y^{2} - 5}, 2 x^{2} - 7 y^{2} + 10 = 0$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{8 y^{2} - 5}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $2 x^{2} - 7 y^{2} + 10 = 0$ with $- \sqrt{8 y^{2} - 5}$ .

$2 {(- \sqrt{8 y^{2} - 5})}^{2} - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $2 {(- \sqrt{8 y^{2} - 5})}^{2} - 7 y^{2} + 10$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{8 y^{2} - 5}$ .

$2 ({(- 1)}^{2} {\sqrt{8 y^{2} - 5}}^{2}) - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.2

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

$2 (1 {\sqrt{8 y^{2} - 5}}^{2}) - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{8 y^{2} - 5}}^{2}$ by $1$ .

$2 {\sqrt{8 y^{2} - 5}}^{2} - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{8 y^{2} - 5}}^{2}$ as $8 y^{2} - 5$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 y^{2} - 5}$ as ${(8 y^{2} - 5)}^{\frac{1}{2}}$ .

$2 {({(8 y^{2} - 5)}^{\frac{1}{2}})}^{2} - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.4.2

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

$2 {(8 y^{2} - 5)}^{\frac{1}{2} \cdot 2} - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.4.3

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

$2 {(8 y^{2} - 5)}^{\frac{2}{2}} - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

$2 {(8 y^{2} - 5)}^{\frac{2}{2}} - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$2 (8 y^{2} - 5) - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

$2 (8 y^{2} - 5) - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.4.5

Simplify.

$2 (8 y^{2} - 5) - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

$2 (8 y^{2} - 5) - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.5

Apply the distributive property.

$2 (8 y^{2}) + 2 \cdot - 5 - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.6

Multiply $8$ by $2$ .

$16 y^{2} + 2 \cdot - 5 - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.1.7

Multiply $2$ by $- 5$ .

$16 y^{2} - 10 - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

$16 y^{2} - 10 - 7 y^{2} + 10 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.2

Simplify by adding terms.

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Step 3.1.2.1.2.1

Combine the opposite terms in $16 y^{2} - 10 - 7 y^{2} + 10$ .

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Step 3.1.2.1.2.1.1

Add $- 10$ and $10$ .

$16 y^{2} - 7 y^{2} + 0 = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.2.1.2

Add $16 y^{2} - 7 y^{2}$ and $0$ .

$16 y^{2} - 7 y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

$16 y^{2} - 7 y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.1.2.1.2.2

Subtract $7 y^{2}$ from $16 y^{2}$ .

$9 y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

$9 y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

$9 y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

$9 y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

$9 y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.2

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

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Step 3.2.1

Divide each term in $9 y^{2} = 0$ by $9$ and simplify.

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Step 3.2.1.1

Divide each term in $9 y^{2} = 0$ by $9$ .

$\frac{9 y^{2}}{9} = \frac{0}{9}$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.2.1.2

Simplify the left side.

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Step 3.2.1.2.1

Cancel the common factor of $9$ .

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Step 3.2.1.2.1.1

Cancel the common factor.

$\frac{9 y^{2}}{9} = \frac{0}{9}$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.2.1.2.1.2

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

$y^{2} = \frac{0}{9}$

$x = - \sqrt{8 y^{2} - 5}$

$y^{2} = \frac{0}{9}$

$x = - \sqrt{8 y^{2} - 5}$

$y^{2} = \frac{0}{9}$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.2.1.3

Simplify the right side.

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Step 3.2.1.3.1

Divide $0$ by $9$ .

$y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

$y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

$y^{2} = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.2.2

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

$y = \pm \sqrt{0}$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.2.3

Simplify $\pm \sqrt{0}$ .

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Step 3.2.3.1

Rewrite $0$ as $0^{2}$ .

$y = \pm \sqrt{0^{2}}$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.2.3.2

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

$y = \pm 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.2.3.3

Plus or minus $0$ is $0$ .

$y = 0$

$x = - \sqrt{8 y^{2} - 5}$

$y = 0$

$x = - \sqrt{8 y^{2} - 5}$

$y = 0$

$x = - \sqrt{8 y^{2} - 5}$

Step 3.3

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

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \sqrt{8 y^{2} - 5}$ with $0$ .

$x = - \sqrt{8 {(0)}^{2} - 5}$

$y = 0$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{8 {(0)}^{2} - 5}$ .

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Step 3.3.2.1.1

Simplify the expression.

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Step 3.3.2.1.1.1

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

$x = - \sqrt{8 \cdot 0 - 5}$

$y = 0$

Step 3.3.2.1.1.2

Multiply $8$ by $0$ .

$x = - \sqrt{0 - 5}$

$y = 0$

Step 3.3.2.1.1.3

Subtract $5$ from $0$ .

$x = - \sqrt{- 5}$

$y = 0$

$x = - \sqrt{- 5}$

$y = 0$

Step 3.3.2.1.2

Rewrite $- 5$ as $- 1 (5)$ .

$x = - \sqrt{- 1 \cdot 5}$

$y = 0$

Step 3.3.2.1.3

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$x = - (\sqrt{- 1} \cdot \sqrt{5})$

$y = 0$

Step 3.3.2.1.4

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

$x = - i \sqrt{5}$

$y = 0$

$x = - i \sqrt{5}$

$y = 0$

$x = - i \sqrt{5}$

$y = 0$

$x = - i \sqrt{5}$

$y = 0$

$x = - i \sqrt{5}$

$y = 0$

Step 4

List all of the solutions.

$x = i \sqrt{5}, y = 0$

$x = - i \sqrt{5}, y = 0$

Step 5