Algebra Examples

$2 x^{2} + 3 y^{2} = 19$ $2 x^{2} + y^{2} = 9$

Step 1

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

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Step 1.1

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

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

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

Step 1.2

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

$y = \pm \sqrt{9 - 2 x^{2}}$

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

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.

$y = \sqrt{9 - 2 x^{2}}$

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

Step 1.3.2

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

$y = - \sqrt{9 - 2 x^{2}}$

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

Step 1.3.3

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

$y = \sqrt{9 - 2 x^{2}}$

$y = - \sqrt{9 - 2 x^{2}}$

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

$y = \sqrt{9 - 2 x^{2}}$

$y = - \sqrt{9 - 2 x^{2}}$

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

$y = \sqrt{9 - 2 x^{2}}$

$y = - \sqrt{9 - 2 x^{2}}$

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

Step 2

Solve the system $y = \sqrt{9 - 2 x^{2}}, 2 x^{2} + 3 y^{2} = 19$ .

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Step 2.1

Replace all occurrences of $y$ with $\sqrt{9 - 2 x^{2}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $y$ in $2 x^{2} + 3 y^{2} = 19$ with $\sqrt{9 - 2 x^{2}}$ .

$2 x^{2} + 3 {(\sqrt{9 - 2 x^{2}})}^{2} = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $2 x^{2} + 3 {(\sqrt{9 - 2 x^{2}})}^{2}$ .

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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 ${\sqrt{9 - 2 x^{2}}}^{2}$ as $9 - 2 x^{2}$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - 2 x^{2}}$ as ${(9 - 2 x^{2})}^{\frac{1}{2}}$ .

$2 x^{2} + 3 {({(9 - 2 x^{2})}^{\frac{1}{2}})}^{2} = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2.1.1.1.2

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

$2 x^{2} + 3 {(9 - 2 x^{2})}^{\frac{1}{2} \cdot 2} = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2.1.1.1.3

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

$2 x^{2} + 3 {(9 - 2 x^{2})}^{\frac{2}{2}} = 19$

$y = \sqrt{9 - 2 x^{2}}$

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.

$2 x^{2} + 3 {(9 - 2 x^{2})}^{\frac{2}{2}} = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$2 x^{2} + 3 (9 - 2 x^{2}) = 19$

$y = \sqrt{9 - 2 x^{2}}$

$2 x^{2} + 3 (9 - 2 x^{2}) = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2.1.1.1.5

Simplify.

$2 x^{2} + 3 (9 - 2 x^{2}) = 19$

$y = \sqrt{9 - 2 x^{2}}$

$2 x^{2} + 3 (9 - 2 x^{2}) = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2.1.1.2

Apply the distributive property.

$2 x^{2} + 3 \cdot 9 + 3 (- 2 x^{2}) = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2.1.1.3

Multiply $3$ by $9$ .

$2 x^{2} + 27 + 3 (- 2 x^{2}) = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2.1.1.4

Multiply $- 2$ by $3$ .

$2 x^{2} + 27 - 6 x^{2} = 19$

$y = \sqrt{9 - 2 x^{2}}$

$2 x^{2} + 27 - 6 x^{2} = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.1.2.1.2

Subtract $6 x^{2}$ from $2 x^{2}$ .

$- 4 x^{2} + 27 = 19$

$y = \sqrt{9 - 2 x^{2}}$

$- 4 x^{2} + 27 = 19$

$y = \sqrt{9 - 2 x^{2}}$

$- 4 x^{2} + 27 = 19$

$y = \sqrt{9 - 2 x^{2}}$

$- 4 x^{2} + 27 = 19$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2

Solve for $x$ in $- 4 x^{2} + 27 = 19$ .

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Step 2.2.1

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

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Step 2.2.1.1

Subtract $27$ from both sides of the equation.

$- 4 x^{2} = 19 - 27$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.1.2

Subtract $27$ from $19$ .

$- 4 x^{2} = - 8$

$y = \sqrt{9 - 2 x^{2}}$

$- 4 x^{2} = - 8$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.2

Divide each term in $- 4 x^{2} = - 8$ by $- 4$ and simplify.

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Step 2.2.2.1

Divide each term in $- 4 x^{2} = - 8$ by $- 4$ .

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

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.2.2.1.2

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

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

$y = \sqrt{9 - 2 x^{2}}$

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

$y = \sqrt{9 - 2 x^{2}}$

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

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $- 8$ by $- 4$ .

$x^{2} = 2$

$y = \sqrt{9 - 2 x^{2}}$

$x^{2} = 2$

$y = \sqrt{9 - 2 x^{2}}$

$x^{2} = 2$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.3

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

$x = \pm \sqrt{2}$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.4

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

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Step 2.2.4.1

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

$x = \sqrt{2}$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.4.2

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

$x = - \sqrt{2}$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.2.4.3

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

$x = \sqrt{2}, - \sqrt{2}$

$y = \sqrt{9 - 2 x^{2}}$

$x = \sqrt{2}, - \sqrt{2}$

$y = \sqrt{9 - 2 x^{2}}$

$x = \sqrt{2}, - \sqrt{2}$

$y = \sqrt{9 - 2 x^{2}}$

Step 2.3

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

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Step 2.3.1

Replace all occurrences of $x$ in $y = \sqrt{9 - 2 x^{2}}$ with $\sqrt{2}$ .

$y = \sqrt{9 - 2 {(\sqrt{2})}^{2}}$

$x = \sqrt{2}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{9 - 2 {(\sqrt{2})}^{2}}$ .

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

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

$x = \sqrt{2}$

Step 2.3.2.1.1.2

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

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

$x = \sqrt{2}$

Step 2.3.2.1.1.3

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

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

$x = \sqrt{2}$

Step 2.3.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.4.1

Cancel the common factor.

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

$x = \sqrt{2}$

Step 2.3.2.1.1.4.2

Rewrite the expression.

$y = \sqrt{9 - 2 \cdot 2}$

$x = \sqrt{2}$

$y = \sqrt{9 - 2 \cdot 2}$

$x = \sqrt{2}$

Step 2.3.2.1.1.5

Evaluate the exponent.

$y = \sqrt{9 - 2 \cdot 2}$

$x = \sqrt{2}$

$y = \sqrt{9 - 2 \cdot 2}$

$x = \sqrt{2}$

Step 2.3.2.1.2

Simplify the expression.

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Step 2.3.2.1.2.1

Multiply $- 2$ by $2$ .

$y = \sqrt{9 - 4}$

$x = \sqrt{2}$

Step 2.3.2.1.2.2

Subtract $4$ from $9$ .

$y = \sqrt{5}$

$x = \sqrt{2}$

$y = \sqrt{5}$

$x = \sqrt{2}$

$y = \sqrt{5}$

$x = \sqrt{2}$

$y = \sqrt{5}$

$x = \sqrt{2}$

$y = \sqrt{5}$

$x = \sqrt{2}$

Step 2.4

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

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Step 2.4.1

Replace all occurrences of $x$ in $y = \sqrt{9 - 2 x^{2}}$ with $- \sqrt{2}$ .

$y = \sqrt{9 - 2 {(- \sqrt{2})}^{2}}$

$x = - \sqrt{2}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{9 - 2 {(- \sqrt{2})}^{2}}$ .

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Step 2.4.2.1.1

Simplify the expression.

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Step 2.4.2.1.1.1

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

$y = \sqrt{9 - 2 ({(- 1)}^{2} {\sqrt{2}}^{2})}$

$x = - \sqrt{2}$

Step 2.4.2.1.1.2

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

$y = \sqrt{9 - 2 (1 {\sqrt{2}}^{2})}$

$x = - \sqrt{2}$

Step 2.4.2.1.1.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$y = \sqrt{9 - 2 {\sqrt{2}}^{2}}$

$x = - \sqrt{2}$

$y = \sqrt{9 - 2 {\sqrt{2}}^{2}}$

$x = - \sqrt{2}$

Step 2.4.2.1.2

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

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Step 2.4.2.1.2.1

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

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

$x = - \sqrt{2}$

Step 2.4.2.1.2.2

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

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

$x = - \sqrt{2}$

Step 2.4.2.1.2.3

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

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

$x = - \sqrt{2}$

Step 2.4.2.1.2.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.2.4.1

Cancel the common factor.

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

$x = - \sqrt{2}$

Step 2.4.2.1.2.4.2

Rewrite the expression.

$y = \sqrt{9 - 2 \cdot 2}$

$x = - \sqrt{2}$

$y = \sqrt{9 - 2 \cdot 2}$

$x = - \sqrt{2}$

Step 2.4.2.1.2.5

Evaluate the exponent.

$y = \sqrt{9 - 2 \cdot 2}$

$x = - \sqrt{2}$

$y = \sqrt{9 - 2 \cdot 2}$

$x = - \sqrt{2}$

Step 2.4.2.1.3

Simplify the expression.

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Step 2.4.2.1.3.1

Multiply $- 2$ by $2$ .

$y = \sqrt{9 - 4}$

$x = - \sqrt{2}$

Step 2.4.2.1.3.2

Subtract $4$ from $9$ .

$y = \sqrt{5}$

$x = - \sqrt{2}$

$y = \sqrt{5}$

$x = - \sqrt{2}$

$y = \sqrt{5}$

$x = - \sqrt{2}$

$y = \sqrt{5}$

$x = - \sqrt{2}$

$y = \sqrt{5}$

$x = - \sqrt{2}$

$y = \sqrt{5}$

$x = - \sqrt{2}$

Step 3

Solve the system $y = - \sqrt{9 - 2 x^{2}}, 2 x^{2} + 3 y^{2} = 19$ .

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Step 3.1

Replace all occurrences of $y$ with $- \sqrt{9 - 2 x^{2}}$ in each equation.

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Step 3.1.1

Replace all occurrences of $y$ in $2 x^{2} + 3 y^{2} = 19$ with $- \sqrt{9 - 2 x^{2}}$ .

$2 x^{2} + 3 {(- \sqrt{9 - 2 x^{2}})}^{2} = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $2 x^{2} + 3 {(- \sqrt{9 - 2 x^{2}})}^{2}$ .

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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 $- \sqrt{9 - 2 x^{2}}$ .

$2 x^{2} + 3 ({(- 1)}^{2} {\sqrt{9 - 2 x^{2}}}^{2}) = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.2

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

$2 x^{2} + 3 (1 {\sqrt{9 - 2 x^{2}}}^{2}) = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{9 - 2 x^{2}}}^{2}$ by $1$ .

$2 x^{2} + 3 {\sqrt{9 - 2 x^{2}}}^{2} = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{9 - 2 x^{2}}}^{2}$ as $9 - 2 x^{2}$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - 2 x^{2}}$ as ${(9 - 2 x^{2})}^{\frac{1}{2}}$ .

$2 x^{2} + 3 {({(9 - 2 x^{2})}^{\frac{1}{2}})}^{2} = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.4.2

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

$2 x^{2} + 3 {(9 - 2 x^{2})}^{\frac{1}{2} \cdot 2} = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.4.3

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

$2 x^{2} + 3 {(9 - 2 x^{2})}^{\frac{2}{2}} = 19$

$y = - \sqrt{9 - 2 x^{2}}$

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.

$2 x^{2} + 3 {(9 - 2 x^{2})}^{\frac{2}{2}} = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$2 x^{2} + 3 (9 - 2 x^{2}) = 19$

$y = - \sqrt{9 - 2 x^{2}}$

$2 x^{2} + 3 (9 - 2 x^{2}) = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.4.5

Simplify.

$2 x^{2} + 3 (9 - 2 x^{2}) = 19$

$y = - \sqrt{9 - 2 x^{2}}$

$2 x^{2} + 3 (9 - 2 x^{2}) = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.5

Apply the distributive property.

$2 x^{2} + 3 \cdot 9 + 3 (- 2 x^{2}) = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.6

Multiply $3$ by $9$ .

$2 x^{2} + 27 + 3 (- 2 x^{2}) = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.1.7

Multiply $- 2$ by $3$ .

$2 x^{2} + 27 - 6 x^{2} = 19$

$y = - \sqrt{9 - 2 x^{2}}$

$2 x^{2} + 27 - 6 x^{2} = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.1.2.1.2

Subtract $6 x^{2}$ from $2 x^{2}$ .

$- 4 x^{2} + 27 = 19$

$y = - \sqrt{9 - 2 x^{2}}$

$- 4 x^{2} + 27 = 19$

$y = - \sqrt{9 - 2 x^{2}}$

$- 4 x^{2} + 27 = 19$

$y = - \sqrt{9 - 2 x^{2}}$

$- 4 x^{2} + 27 = 19$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2

Solve for $x$ in $- 4 x^{2} + 27 = 19$ .

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Step 3.2.1

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

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Step 3.2.1.1

Subtract $27$ from both sides of the equation.

$- 4 x^{2} = 19 - 27$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.1.2

Subtract $27$ from $19$ .

$- 4 x^{2} = - 8$

$y = - \sqrt{9 - 2 x^{2}}$

$- 4 x^{2} = - 8$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.2

Divide each term in $- 4 x^{2} = - 8$ by $- 4$ and simplify.

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Step 3.2.2.1

Divide each term in $- 4 x^{2} = - 8$ by $- 4$ .

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

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.2.2.1.2

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

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

$y = - \sqrt{9 - 2 x^{2}}$

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

$y = - \sqrt{9 - 2 x^{2}}$

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

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 8$ by $- 4$ .

$x^{2} = 2$

$y = - \sqrt{9 - 2 x^{2}}$

$x^{2} = 2$

$y = - \sqrt{9 - 2 x^{2}}$

$x^{2} = 2$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.3

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

$x = \pm \sqrt{2}$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.4

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

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Step 3.2.4.1

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

$x = \sqrt{2}$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.4.2

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

$x = - \sqrt{2}$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.2.4.3

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

$x = \sqrt{2}, - \sqrt{2}$

$y = - \sqrt{9 - 2 x^{2}}$

$x = \sqrt{2}, - \sqrt{2}$

$y = - \sqrt{9 - 2 x^{2}}$

$x = \sqrt{2}, - \sqrt{2}$

$y = - \sqrt{9 - 2 x^{2}}$

Step 3.3

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

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Step 3.3.1

Replace all occurrences of $x$ in $y = - \sqrt{9 - 2 x^{2}}$ with $\sqrt{2}$ .

$y = - \sqrt{9 - 2 {(\sqrt{2})}^{2}}$

$x = \sqrt{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{9 - 2 {(\sqrt{2})}^{2}}$ .

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Step 3.3.2.1.1

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

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Step 3.3.2.1.1.1

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

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

$x = \sqrt{2}$

Step 3.3.2.1.1.2

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

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

$x = \sqrt{2}$

Step 3.3.2.1.1.3

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

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

$x = \sqrt{2}$

Step 3.3.2.1.1.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.4.1

Cancel the common factor.

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

$x = \sqrt{2}$

Step 3.3.2.1.1.4.2

Rewrite the expression.

$y = - \sqrt{9 - 2 \cdot 2}$

$x = \sqrt{2}$

$y = - \sqrt{9 - 2 \cdot 2}$

$x = \sqrt{2}$

Step 3.3.2.1.1.5

Evaluate the exponent.

$y = - \sqrt{9 - 2 \cdot 2}$

$x = \sqrt{2}$

$y = - \sqrt{9 - 2 \cdot 2}$

$x = \sqrt{2}$

Step 3.3.2.1.2

Simplify the expression.

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Step 3.3.2.1.2.1

Multiply $- 2$ by $2$ .

$y = - \sqrt{9 - 4}$

$x = \sqrt{2}$

Step 3.3.2.1.2.2

Subtract $4$ from $9$ .

$y = - \sqrt{5}$

$x = \sqrt{2}$

$y = - \sqrt{5}$

$x = \sqrt{2}$

$y = - \sqrt{5}$

$x = \sqrt{2}$

$y = - \sqrt{5}$

$x = \sqrt{2}$

$y = - \sqrt{5}$

$x = \sqrt{2}$

Step 3.4

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

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Step 3.4.1

Replace all occurrences of $x$ in $y = - \sqrt{9 - 2 x^{2}}$ with $- \sqrt{2}$ .

$y = - \sqrt{9 - 2 {(- \sqrt{2})}^{2}}$

$x = - \sqrt{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{9 - 2 {(- \sqrt{2})}^{2}}$ .

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Step 3.4.2.1.1

Simplify the expression.

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Step 3.4.2.1.1.1

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

$y = - \sqrt{9 - 2 ({(- 1)}^{2} {\sqrt{2}}^{2})}$

$x = - \sqrt{2}$

Step 3.4.2.1.1.2

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

$y = - \sqrt{9 - 2 (1 {\sqrt{2}}^{2})}$

$x = - \sqrt{2}$

Step 3.4.2.1.1.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$y = - \sqrt{9 - 2 {\sqrt{2}}^{2}}$

$x = - \sqrt{2}$

$y = - \sqrt{9 - 2 {\sqrt{2}}^{2}}$

$x = - \sqrt{2}$

Step 3.4.2.1.2

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

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Step 3.4.2.1.2.1

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

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

$x = - \sqrt{2}$

Step 3.4.2.1.2.2

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

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

$x = - \sqrt{2}$

Step 3.4.2.1.2.3

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

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

$x = - \sqrt{2}$

Step 3.4.2.1.2.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.2.4.1

Cancel the common factor.

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

$x = - \sqrt{2}$

Step 3.4.2.1.2.4.2

Rewrite the expression.

$y = - \sqrt{9 - 2 \cdot 2}$

$x = - \sqrt{2}$

$y = - \sqrt{9 - 2 \cdot 2}$

$x = - \sqrt{2}$

Step 3.4.2.1.2.5

Evaluate the exponent.

$y = - \sqrt{9 - 2 \cdot 2}$

$x = - \sqrt{2}$

$y = - \sqrt{9 - 2 \cdot 2}$

$x = - \sqrt{2}$

Step 3.4.2.1.3

Simplify the expression.

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Step 3.4.2.1.3.1

Multiply $- 2$ by $2$ .

$y = - \sqrt{9 - 4}$

$x = - \sqrt{2}$

Step 3.4.2.1.3.2

Subtract $4$ from $9$ .

$y = - \sqrt{5}$

$x = - \sqrt{2}$

$y = - \sqrt{5}$

$x = - \sqrt{2}$

$y = - \sqrt{5}$

$x = - \sqrt{2}$

$y = - \sqrt{5}$

$x = - \sqrt{2}$

$y = - \sqrt{5}$

$x = - \sqrt{2}$

$y = - \sqrt{5}$

$x = - \sqrt{2}$

Step 4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\sqrt{2}, \sqrt{5})$

$(- \sqrt{2}, \sqrt{5})$

$(\sqrt{2}, - \sqrt{5})$

$(- \sqrt{2}, - \sqrt{5})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(\sqrt{2}, \sqrt{5}), (- \sqrt{2}, \sqrt{5}), (\sqrt{2}, - \sqrt{5}), (- \sqrt{2}, - \sqrt{5})$

Equation Form:

$x = \sqrt{2}, y = \sqrt{5}$

$x = - \sqrt{2}, y = \sqrt{5}$

$x = \sqrt{2}, y = - \sqrt{5}$

$x = - \sqrt{2}, y = - \sqrt{5}$

Step 6