Finite Math Examples

$3 x^{2} - 2 y^{2} + 5 = 0$ , $2 x^{2} - y^{2} + 2 = 0$

Step 1

Solve for $x$ in $3 x^{2} - 2 y^{2} + 5 = 0$ .

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

$3 x^{2} + 5 = 2 y^{2}$

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

Step 1.1.2

Subtract $5$ from both sides of the equation.

$3 x^{2} = 2 y^{2} - 5$

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

$3 x^{2} = 2 y^{2} - 5$

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

Step 1.2

Divide each term in $3 x^{2} = 2 y^{2} - 5$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x^{2} = 2 y^{2} - 5$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{2 y^{2}}{3} + \frac{- 5}{3}$

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{2 y^{2}}{3} + \frac{- 5}{3}$

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

Step 1.2.2.1.2

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

$x^{2} = \frac{2 y^{2}}{3} + \frac{- 5}{3}$

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

$x^{2} = \frac{2 y^{2}}{3} + \frac{- 5}{3}$

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

$x^{2} = \frac{2 y^{2}}{3} + \frac{- 5}{3}$

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x^{2} = \frac{2 y^{2}}{3} - \frac{5}{3}$

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

$x^{2} = \frac{2 y^{2}}{3} - \frac{5}{3}$

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

$x^{2} = \frac{2 y^{2}}{3} - \frac{5}{3}$

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

Step 1.3

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

$x = \pm \sqrt{\frac{2 y^{2}}{3} - \frac{5}{3}}$

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

Step 1.4

Simplify $\pm \sqrt{\frac{2 y^{2}}{3} - \frac{5}{3}}$ .

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Step 1.4.1

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{2 y^{2} - 5}{3}}$

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

Step 1.4.2

Rewrite $\sqrt{\frac{2 y^{2} - 5}{3}}$ as $\frac{\sqrt{2 y^{2} - 5}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2 y^{2} - 5}}{\sqrt{3}}$

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

Step 1.4.3

Multiply $\frac{\sqrt{2 y^{2} - 5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2 y^{2} - 5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

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

Step 1.4.4

Combine and simplify the denominator.

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Step 1.4.4.1

Multiply $\frac{\sqrt{2 y^{2} - 5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

Step 1.4.4.2

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

Step 1.4.4.3

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

Step 1.4.4.4

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

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

Step 1.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{{\sqrt{3}}^{2}}$

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

Step 1.4.4.6

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

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Step 1.4.4.6.1

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

Step 1.4.4.6.2

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

Step 1.4.4.6.3

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{3^{\frac{2}{2}}}$

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

Step 1.4.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{3^{\frac{2}{2}}}$

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

Step 1.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{3}$

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{3}$

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

Step 1.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{3}$

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{3}$

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

$x = \pm \frac{\sqrt{2 y^{2} - 5} \sqrt{3}}{3}$

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

Step 1.4.5

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(2 y^{2} - 5) \cdot 3}}{3}$

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

Step 1.4.6

Reorder factors in $\pm \frac{\sqrt{(2 y^{2} - 5) \cdot 3}}{3}$ .

$x = \pm \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = \pm \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

Step 1.5

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

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Step 1.5.1

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

Step 1.5.2

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

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

Step 1.5.3

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

Step 2

Solve the system $x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}, 2 x^{2} - y^{2} + 2 = 0$ .

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Step 2.1

Replace all occurrences of $x$ with $\frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $2 x^{2} - y^{2} + 2 = 0$ with $\frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ .

$2 {(\frac{\sqrt{3 (2 y^{2} - 5)}}{3})}^{2} - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $2 {(\frac{\sqrt{3 (2 y^{2} - 5)}}{3})}^{2} - y^{2} + 2$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Apply the product rule to $\frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ .

$2 (\frac{{\sqrt{3 (2 y^{2} - 5)}}^{2}}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2

Simplify the numerator.

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Step 2.1.2.1.1.2.1

Rewrite ${\sqrt{3 (2 y^{2} - 5)}}^{2}$ as $3 (2 y^{2} - 5)$ .

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Step 2.1.2.1.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (2 y^{2} - 5)}$ as ${(3 (2 y^{2} - 5))}^{\frac{1}{2}}$ .

$2 (\frac{{({(3 (2 y^{2} - 5))}^{\frac{1}{2}})}^{2}}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.1.2

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

$2 (\frac{{(3 (2 y^{2} - 5))}^{\frac{1}{2} \cdot 2}}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.1.3

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

$2 (\frac{{(3 (2 y^{2} - 5))}^{\frac{2}{2}}}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.2.1.4.1

Cancel the common factor.

$2 (\frac{{(3 (2 y^{2} - 5))}^{\frac{2}{2}}}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.1.4.2

Rewrite the expression.

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.1.5

Simplify.

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$2 (\frac{3 (2 y^{2}) + 3 \cdot - 5}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.3

Multiply $2$ by $3$ .

$2 (\frac{6 y^{2} + 3 \cdot - 5}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.4

Multiply $3$ by $- 5$ .

$2 (\frac{6 y^{2} - 15}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.5

Factor $3$ out of $6 y^{2} - 15$ .

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Step 2.1.2.1.1.2.5.1

Factor $3$ out of $6 y^{2}$ .

$2 (\frac{3 (2 y^{2}) - 15}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.5.2

Factor $3$ out of $- 15$ .

$2 (\frac{3 (2 y^{2}) + 3 (- 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.2.5.3

Factor $3$ out of $3 (2 y^{2}) + 3 (- 5)$ .

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.3

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

$2 (\frac{3 (2 y^{2} - 5)}{9}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.4

Cancel the common factors.

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Step 2.1.2.1.1.4.1

Factor $3$ out of $9$ .

$2 (\frac{3 (2 y^{2} - 5)}{3 \cdot 3}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.4.2

Cancel the common factor.

$2 (\frac{3 (2 y^{2} - 5)}{3 \cdot 3}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.4.3

Rewrite the expression.

$2 (\frac{2 y^{2} - 5}{3}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{2 y^{2} - 5}{3}) - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.1.5

Combine $2$ and $\frac{2 y^{2} - 5}{3}$ .

$\frac{2 (2 y^{2} - 5)}{3} - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$\frac{2 (2 y^{2} - 5)}{3} - y^{2} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.2

To write $- y^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 (2 y^{2} - 5)}{3} - y^{2} \cdot \frac{3}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.3

Simplify terms.

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Step 2.1.2.1.3.1

Combine $- y^{2}$ and $\frac{3}{3}$ .

$\frac{2 (2 y^{2} - 5)}{3} + \frac{- y^{2} \cdot 3}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.3.2

Combine the numerators over the common denominator.

$\frac{2 (2 y^{2} - 5) - y^{2} \cdot 3}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$\frac{2 (2 y^{2} - 5) - y^{2} \cdot 3}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.4

Simplify the numerator.

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Step 2.1.2.1.4.1

Apply the distributive property.

$\frac{2 (2 y^{2}) + 2 \cdot - 5 - y^{2} \cdot 3}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.4.2

Multiply $2$ by $2$ .

$\frac{4 y^{2} + 2 \cdot - 5 - y^{2} \cdot 3}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.4.3

Multiply $2$ by $- 5$ .

$\frac{4 y^{2} - 10 - y^{2} \cdot 3}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.4.4

Multiply $3$ by $- 1$ .

$\frac{4 y^{2} - 10 - 3 y^{2}}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.4.5

Subtract $3 y^{2}$ from $4 y^{2}$ .

$\frac{y^{2} - 10}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$\frac{y^{2} - 10}{3} + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{y^{2} - 10}{3} + 2 \cdot \frac{3}{3} = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.6

Simplify terms.

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Step 2.1.2.1.6.1

Combine $2$ and $\frac{3}{3}$ .

$\frac{y^{2} - 10}{3} + \frac{2 \cdot 3}{3} = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.6.2

Combine the numerators over the common denominator.

$\frac{y^{2} - 10 + 2 \cdot 3}{3} = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$\frac{y^{2} - 10 + 2 \cdot 3}{3} = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.7

Simplify the numerator.

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Step 2.1.2.1.7.1

Multiply $2$ by $3$ .

$\frac{y^{2} - 10 + 6}{3} = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.7.2

Add $- 10$ and $6$ .

$\frac{y^{2} - 4}{3} = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.7.3

Rewrite $y^{2} - 4$ in a factored form.

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Step 2.1.2.1.7.3.1

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

$\frac{y^{2} - 2^{2}}{3} = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.1.2.1.7.3.2

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 $b = 2$ .

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.2

Solve for $y$ in $\frac{(y + 2) (y - 2)}{3} = 0$ .

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Step 2.2.1

Set the numerator equal to zero.

$(y + 2) (y - 2) = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.2.2

Solve the equation for $y$ .

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Step 2.2.2.1

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

$y + 2 = 0$

$y - 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.2.2.2

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

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Step 2.2.2.2.1

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

$y + 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.2.2.2.2

Subtract $2$ from both sides of the equation.

$y = - 2$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$y = - 2$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.2.2.3

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

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Step 2.2.2.3.1

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

$y - 2 = 0$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.2.2.3.2

Add $2$ to both sides of the equation.

$y = 2$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$y = 2$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.2.2.4

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

$y = - 2, 2$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$y = - 2, 2$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$y = - 2, 2$

$x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 2.3

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

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Step 2.3.1

Replace all occurrences of $y$ in $x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ with $- 2$ .

$x = \frac{\sqrt{3 (2 {(- 2)}^{2} - 5)}}{3}$

$y = - 2$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\frac{\sqrt{3 (2 {(- 2)}^{2} - 5)}}{3}$ .

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Step 2.3.2.1.1

Simplify the numerator.

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Step 2.3.2.1.1.1

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

$x = \frac{\sqrt{3 (2 \cdot 4 - 5)}}{3}$

$y = - 2$

Step 2.3.2.1.1.2

Multiply $2$ by $4$ .

$x = \frac{\sqrt{3 (8 - 5)}}{3}$

$y = - 2$

Step 2.3.2.1.1.3

Subtract $5$ from $8$ .

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

$y = - 2$

Step 2.3.2.1.1.4

Multiply $3$ by $3$ .

$x = \frac{\sqrt{9}}{3}$

$y = - 2$

Step 2.3.2.1.1.5

Rewrite $9$ as $3^{2}$ .

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

$y = - 2$

Step 2.3.2.1.1.6

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

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

$y = - 2$

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

$y = - 2$

Step 2.3.2.1.2

Divide $3$ by $3$ .

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

Step 2.4

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

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Step 2.4.1

Replace all occurrences of $y$ in $x = \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ with $2$ .

$x = \frac{\sqrt{3 (2 {(2)}^{2} - 5)}}{3}$

$y = 2$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\frac{\sqrt{3 (2 {(2)}^{2} - 5)}}{3}$ .

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Step 2.4.2.1.1

Simplify the numerator.

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Step 2.4.2.1.1.1

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 2.4.2.1.1.1.1

Multiply $2$ by $2^{2}$ .

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Step 2.4.2.1.1.1.1.1

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

$x = \frac{\sqrt{3 (2 \cdot 2^{2} - 5)}}{3}$

$y = 2$

Step 2.4.2.1.1.1.1.2

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

$x = \frac{\sqrt{3 (2^{1 + 2} - 5)}}{3}$

$y = 2$

$x = \frac{\sqrt{3 (2^{1 + 2} - 5)}}{3}$

$y = 2$

Step 2.4.2.1.1.1.2

Add $1$ and $2$ .

$x = \frac{\sqrt{3 (2^{3} - 5)}}{3}$

$y = 2$

$x = \frac{\sqrt{3 (2^{3} - 5)}}{3}$

$y = 2$

Step 2.4.2.1.1.2

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

$x = \frac{\sqrt{3 (8 - 5)}}{3}$

$y = 2$

Step 2.4.2.1.1.3

Subtract $5$ from $8$ .

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

$y = 2$

Step 2.4.2.1.1.4

Multiply $3$ by $3$ .

$x = \frac{\sqrt{9}}{3}$

$y = 2$

Step 2.4.2.1.1.5

Rewrite $9$ as $3^{2}$ .

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

$y = 2$

Step 2.4.2.1.1.6

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

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

$y = 2$

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

$y = 2$

Step 2.4.2.1.2

Divide $3$ by $3$ .

$x = 1$

$y = 2$

$x = 1$

$y = 2$

$x = 1$

$y = 2$

$x = 1$

$y = 2$

$x = 1$

$y = 2$

Step 3

Solve the system $x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}, 2 x^{2} - y^{2} + 2 = 0$ .

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Step 3.1

Replace all occurrences of $x$ with $- \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $2 x^{2} - y^{2} + 2 = 0$ with $- \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ .

$2 {(- \frac{\sqrt{3 (2 y^{2} - 5)}}{3})}^{2} - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $2 {(- \frac{\sqrt{3 (2 y^{2} - 5)}}{3})}^{2} - y^{2} + 2$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

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

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Step 3.1.2.1.1.1.1

Apply the product rule to $- \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ .

$2 ({(- 1)}^{2} {(\frac{\sqrt{3 (2 y^{2} - 5)}}{3})}^{2}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.1.2

Apply the product rule to $\frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ .

$2 ({(- 1)}^{2} (\frac{{\sqrt{3 (2 y^{2} - 5)}}^{2}}{3^{2}})) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 ({(- 1)}^{2} (\frac{{\sqrt{3 (2 y^{2} - 5)}}^{2}}{3^{2}})) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.2

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

$2 (1 (\frac{{\sqrt{3 (2 y^{2} - 5)}}^{2}}{3^{2}})) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.3

Multiply $\frac{{\sqrt{3 (2 y^{2} - 5)}}^{2}}{3^{2}}$ by $1$ .

$2 (\frac{{\sqrt{3 (2 y^{2} - 5)}}^{2}}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4

Simplify the numerator.

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Step 3.1.2.1.1.4.1

Rewrite ${\sqrt{3 (2 y^{2} - 5)}}^{2}$ as $3 (2 y^{2} - 5)$ .

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Step 3.1.2.1.1.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (2 y^{2} - 5)}$ as ${(3 (2 y^{2} - 5))}^{\frac{1}{2}}$ .

$2 (\frac{{({(3 (2 y^{2} - 5))}^{\frac{1}{2}})}^{2}}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.1.2

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

$2 (\frac{{(3 (2 y^{2} - 5))}^{\frac{1}{2} \cdot 2}}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.1.3

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

$2 (\frac{{(3 (2 y^{2} - 5))}^{\frac{2}{2}}}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.1.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.1.4.1

Cancel the common factor.

$2 (\frac{{(3 (2 y^{2} - 5))}^{\frac{2}{2}}}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.1.4.2

Rewrite the expression.

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.1.5

Simplify.

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.2

Apply the distributive property.

$2 (\frac{3 (2 y^{2}) + 3 \cdot - 5}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.3

Multiply $2$ by $3$ .

$2 (\frac{6 y^{2} + 3 \cdot - 5}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.4

Multiply $3$ by $- 5$ .

$2 (\frac{6 y^{2} - 15}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.5

Factor $3$ out of $6 y^{2} - 15$ .

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Step 3.1.2.1.1.4.5.1

Factor $3$ out of $6 y^{2}$ .

$2 (\frac{3 (2 y^{2}) - 15}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.5.2

Factor $3$ out of $- 15$ .

$2 (\frac{3 (2 y^{2}) + 3 (- 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.4.5.3

Factor $3$ out of $3 (2 y^{2}) + 3 (- 5)$ .

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{3 (2 y^{2} - 5)}{3^{2}}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.5

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

$2 (\frac{3 (2 y^{2} - 5)}{9}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.6

Cancel the common factors.

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Step 3.1.2.1.1.6.1

Factor $3$ out of $9$ .

$2 (\frac{3 (2 y^{2} - 5)}{3 \cdot 3}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.6.2

Cancel the common factor.

$2 (\frac{3 (2 y^{2} - 5)}{3 \cdot 3}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.6.3

Rewrite the expression.

$2 (\frac{2 y^{2} - 5}{3}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$2 (\frac{2 y^{2} - 5}{3}) - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.1.7

Combine $2$ and $\frac{2 y^{2} - 5}{3}$ .

$\frac{2 (2 y^{2} - 5)}{3} - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$\frac{2 (2 y^{2} - 5)}{3} - y^{2} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.2

To write $- y^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 (2 y^{2} - 5)}{3} - y^{2} \cdot \frac{3}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.3

Simplify terms.

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Step 3.1.2.1.3.1

Combine $- y^{2}$ and $\frac{3}{3}$ .

$\frac{2 (2 y^{2} - 5)}{3} + \frac{- y^{2} \cdot 3}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.3.2

Combine the numerators over the common denominator.

$\frac{2 (2 y^{2} - 5) - y^{2} \cdot 3}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$\frac{2 (2 y^{2} - 5) - y^{2} \cdot 3}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.4

Simplify the numerator.

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Step 3.1.2.1.4.1

Apply the distributive property.

$\frac{2 (2 y^{2}) + 2 \cdot - 5 - y^{2} \cdot 3}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.4.2

Multiply $2$ by $2$ .

$\frac{4 y^{2} + 2 \cdot - 5 - y^{2} \cdot 3}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.4.3

Multiply $2$ by $- 5$ .

$\frac{4 y^{2} - 10 - y^{2} \cdot 3}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.4.4

Multiply $3$ by $- 1$ .

$\frac{4 y^{2} - 10 - 3 y^{2}}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.4.5

Subtract $3 y^{2}$ from $4 y^{2}$ .

$\frac{y^{2} - 10}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$\frac{y^{2} - 10}{3} + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{y^{2} - 10}{3} + 2 \cdot \frac{3}{3} = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.6

Simplify terms.

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Step 3.1.2.1.6.1

Combine $2$ and $\frac{3}{3}$ .

$\frac{y^{2} - 10}{3} + \frac{2 \cdot 3}{3} = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.6.2

Combine the numerators over the common denominator.

$\frac{y^{2} - 10 + 2 \cdot 3}{3} = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$\frac{y^{2} - 10 + 2 \cdot 3}{3} = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.7

Simplify the numerator.

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Step 3.1.2.1.7.1

Multiply $2$ by $3$ .

$\frac{y^{2} - 10 + 6}{3} = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.7.2

Add $- 10$ and $6$ .

$\frac{y^{2} - 4}{3} = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.7.3

Rewrite $y^{2} - 4$ in a factored form.

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Step 3.1.2.1.7.3.1

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

$\frac{y^{2} - 2^{2}}{3} = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.1.2.1.7.3.2

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 $b = 2$ .

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

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

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

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.2

Solve for $y$ in $\frac{(y + 2) (y - 2)}{3} = 0$ .

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Step 3.2.1

Set the numerator equal to zero.

$(y + 2) (y - 2) = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.2.2

Solve the equation for $y$ .

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Step 3.2.2.1

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

$y + 2 = 0$

$y - 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.2.2.2

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

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Step 3.2.2.2.1

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

$y + 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.2.2.2.2

Subtract $2$ from both sides of the equation.

$y = - 2$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$y = - 2$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.2.2.3

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

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Step 3.2.2.3.1

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

$y - 2 = 0$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.2.2.3.2

Add $2$ to both sides of the equation.

$y = 2$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$y = 2$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.2.2.4

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

$y = - 2, 2$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$y = - 2, 2$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

$y = - 2, 2$

$x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$

Step 3.3

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

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ with $- 2$ .

$x = - \frac{\sqrt{3 (2 {(- 2)}^{2} - 5)}}{3}$

$y = - 2$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{\sqrt{3 (2 {(- 2)}^{2} - 5)}}{3}$ .

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Step 3.3.2.1.1

Simplify the numerator.

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Step 3.3.2.1.1.1

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

$x = - \frac{\sqrt{3 (2 \cdot 4 - 5)}}{3}$

$y = - 2$

Step 3.3.2.1.1.2

Multiply $2$ by $4$ .

$x = - \frac{\sqrt{3 (8 - 5)}}{3}$

$y = - 2$

Step 3.3.2.1.1.3

Subtract $5$ from $8$ .

$x = - \frac{\sqrt{3 \cdot 3}}{3}$

$y = - 2$

Step 3.3.2.1.1.4

Multiply $3$ by $3$ .

$x = - \frac{\sqrt{9}}{3}$

$y = - 2$

Step 3.3.2.1.1.5

Rewrite $9$ as $3^{2}$ .

$x = - \frac{\sqrt{3^{2}}}{3}$

$y = - 2$

Step 3.3.2.1.1.6

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

$x = - \frac{3}{3}$

$y = - 2$

$x = - \frac{3}{3}$

$y = - 2$

Step 3.3.2.1.2

Reduce the expression by cancelling the common factors.

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Step 3.3.2.1.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.1.1

Cancel the common factor.

$x = - \frac{3}{3}$

$y = - 2$

Step 3.3.2.1.2.1.2

Rewrite the expression.

$x = - 1 \cdot 1$

$y = - 2$

$x = - 1 \cdot 1$

$y = - 2$

Step 3.3.2.1.2.2

Multiply $- 1$ by $1$ .

$x = - 1$

$y = - 2$

$x = - 1$

$y = - 2$

$x = - 1$

$y = - 2$

$x = - 1$

$y = - 2$

$x = - 1$

$y = - 2$

Step 3.4

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

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Step 3.4.1

Replace all occurrences of $y$ in $x = - \frac{\sqrt{3 (2 y^{2} - 5)}}{3}$ with $2$ .

$x = - \frac{\sqrt{3 (2 {(2)}^{2} - 5)}}{3}$

$y = 2$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \frac{\sqrt{3 (2 {(2)}^{2} - 5)}}{3}$ .

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Step 3.4.2.1.1

Simplify the numerator.

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Step 3.4.2.1.1.1

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 3.4.2.1.1.1.1

Multiply $2$ by $2^{2}$ .

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Step 3.4.2.1.1.1.1.1

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

$x = - \frac{\sqrt{3 (2 \cdot 2^{2} - 5)}}{3}$

$y = 2$

Step 3.4.2.1.1.1.1.2

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

$x = - \frac{\sqrt{3 (2^{1 + 2} - 5)}}{3}$

$y = 2$

$x = - \frac{\sqrt{3 (2^{1 + 2} - 5)}}{3}$

$y = 2$

Step 3.4.2.1.1.1.2

Add $1$ and $2$ .

$x = - \frac{\sqrt{3 (2^{3} - 5)}}{3}$

$y = 2$

$x = - \frac{\sqrt{3 (2^{3} - 5)}}{3}$

$y = 2$

Step 3.4.2.1.1.2

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

$x = - \frac{\sqrt{3 (8 - 5)}}{3}$

$y = 2$

Step 3.4.2.1.1.3

Subtract $5$ from $8$ .

$x = - \frac{\sqrt{3 \cdot 3}}{3}$

$y = 2$

Step 3.4.2.1.1.4

Multiply $3$ by $3$ .

$x = - \frac{\sqrt{9}}{3}$

$y = 2$

Step 3.4.2.1.1.5

Rewrite $9$ as $3^{2}$ .

$x = - \frac{\sqrt{3^{2}}}{3}$

$y = 2$

Step 3.4.2.1.1.6

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

$x = - \frac{3}{3}$

$y = 2$

$x = - \frac{3}{3}$

$y = 2$

Step 3.4.2.1.2

Reduce the expression by cancelling the common factors.

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Step 3.4.2.1.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.1.2.1.1

Cancel the common factor.

$x = - \frac{3}{3}$

$y = 2$

Step 3.4.2.1.2.1.2

Rewrite the expression.

$x = - 1 \cdot 1$

$y = 2$

$x = - 1 \cdot 1$

$y = 2$

Step 3.4.2.1.2.2

Multiply $- 1$ by $1$ .

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

$x = - 1$

$y = 2$

Step 4

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

$(1, - 2)$

$(1, 2)$

$(- 1, - 2)$

$(- 1, 2)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(1, - 2), (1, 2), (- 1, - 2), (- 1, 2)$

Equation Form:

$x = 1, y = - 2$

$x = 1, y = 2$

$x = - 1, y = - 2$

$x = - 1, y = 2$

Step 6