Precalculus Examples

$7 x^{2} + 2 y^{2} = 5$ , $6 x^{2} - y^{2} = 7$

Step 1

Solve for $y$ in $6 x^{2} - y^{2} = 7$ .

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Step 1.1

Subtract $6 x^{2}$ from both sides of the equation.

$- y^{2} = 7 - 6 x^{2}$

$7 x^{2} + 2 y^{2} = 5$

Step 1.2

Divide each term in $- y^{2} = 7 - 6 x^{2}$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- y^{2} = 7 - 6 x^{2}$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{7}{- 1} + \frac{- 6 x^{2}}{- 1}$

$7 x^{2} + 2 y^{2} = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{7}{- 1} + \frac{- 6 x^{2}}{- 1}$

$7 x^{2} + 2 y^{2} = 5$

Step 1.2.2.2

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

$y^{2} = \frac{7}{- 1} + \frac{- 6 x^{2}}{- 1}$

$7 x^{2} + 2 y^{2} = 5$

$y^{2} = \frac{7}{- 1} + \frac{- 6 x^{2}}{- 1}$

$7 x^{2} + 2 y^{2} = 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $7$ by $- 1$ .

$y^{2} = - 7 + \frac{- 6 x^{2}}{- 1}$

$7 x^{2} + 2 y^{2} = 5$

Step 1.2.3.1.2

Move the negative one from the denominator of $\frac{- 6 x^{2}}{- 1}$ .

$y^{2} = - 7 - 1 \cdot (- 6 x^{2})$

$7 x^{2} + 2 y^{2} = 5$

Step 1.2.3.1.3

Rewrite $- 1 \cdot (- 6 x^{2})$ as $- (- 6 x^{2})$ .

$y^{2} = - 7 - (- 6 x^{2})$

$7 x^{2} + 2 y^{2} = 5$

Step 1.2.3.1.4

Multiply $- 6$ by $- 1$ .

$y^{2} = - 7 + 6 x^{2}$

$7 x^{2} + 2 y^{2} = 5$

$y^{2} = - 7 + 6 x^{2}$

$7 x^{2} + 2 y^{2} = 5$

$y^{2} = - 7 + 6 x^{2}$

$7 x^{2} + 2 y^{2} = 5$

$y^{2} = - 7 + 6 x^{2}$

$7 x^{2} + 2 y^{2} = 5$

Step 1.3

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

$y = \pm \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 y^{2} = 5$

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.

$y = \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 y^{2} = 5$

Step 1.4.2

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

$y = - \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 y^{2} = 5$

Step 1.4.3

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

$y = \sqrt{- 7 + 6 x^{2}}$

$y = - \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 y^{2} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

$y = - \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 y^{2} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

$y = - \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 y^{2} = 5$

Step 2

Solve the system $y = \sqrt{- 7 + 6 x^{2}}, 7 x^{2} + 2 y^{2} = 5$ .

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Step 2.1

Replace all occurrences of $y$ with $\sqrt{- 7 + 6 x^{2}}$ in each equation.

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Step 2.1.1

Replace all occurrences of $y$ in $7 x^{2} + 2 y^{2} = 5$ with $\sqrt{- 7 + 6 x^{2}}$ .

$7 x^{2} + 2 {(\sqrt{- 7 + 6 x^{2}})}^{2} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $7 x^{2} + 2 {(\sqrt{- 7 + 6 x^{2}})}^{2}$ .

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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 ${\sqrt{- 7 + 6 x^{2}}}^{2}$ as $- 7 + 6 x^{2}$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 7 + 6 x^{2}}$ as ${(- 7 + 6 x^{2})}^{\frac{1}{2}}$ .

$7 x^{2} + 2 {({(- 7 + 6 x^{2})}^{\frac{1}{2}})}^{2} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2.1.1.1.2

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

$7 x^{2} + 2 {(- 7 + 6 x^{2})}^{\frac{1}{2} \cdot 2} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2.1.1.1.3

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

$7 x^{2} + 2 {(- 7 + 6 x^{2})}^{\frac{2}{2}} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

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.

$7 x^{2} + 2 {(- 7 + 6 x^{2})}^{\frac{2}{2}} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$7 x^{2} + 2 (- 7 + 6 x^{2}) = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 (- 7 + 6 x^{2}) = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2.1.1.1.5

Simplify.

$7 x^{2} + 2 (- 7 + 6 x^{2}) = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 (- 7 + 6 x^{2}) = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2.1.1.2

Apply the distributive property.

$7 x^{2} + 2 \cdot - 7 + 2 (6 x^{2}) = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2.1.1.3

Multiply $2$ by $- 7$ .

$7 x^{2} - 14 + 2 (6 x^{2}) = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2.1.1.4

Multiply $6$ by $2$ .

$7 x^{2} - 14 + 12 x^{2} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} - 14 + 12 x^{2} = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.1.2.1.2

Add $7 x^{2}$ and $12 x^{2}$ .

$19 x^{2} - 14 = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

$19 x^{2} - 14 = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

$19 x^{2} - 14 = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

$19 x^{2} - 14 = 5$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2

Solve for $x$ in $19 x^{2} - 14 = 5$ .

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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

Add $14$ to both sides of the equation.

$19 x^{2} = 5 + 14$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.1.2

Add $5$ and $14$ .

$19 x^{2} = 19$

$y = \sqrt{- 7 + 6 x^{2}}$

$19 x^{2} = 19$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.2

Divide each term in $19 x^{2} = 19$ by $19$ and simplify.

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Step 2.2.2.1

Divide each term in $19 x^{2} = 19$ by $19$ .

$\frac{19 x^{2}}{19} = \frac{19}{19}$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $19$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{19 x^{2}}{19} = \frac{19}{19}$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.2.2.1.2

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

$x^{2} = \frac{19}{19}$

$y = \sqrt{- 7 + 6 x^{2}}$

$x^{2} = \frac{19}{19}$

$y = \sqrt{- 7 + 6 x^{2}}$

$x^{2} = \frac{19}{19}$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $19$ by $19$ .

$x^{2} = 1$

$y = \sqrt{- 7 + 6 x^{2}}$

$x^{2} = 1$

$y = \sqrt{- 7 + 6 x^{2}}$

$x^{2} = 1$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.3

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

$x = \pm \sqrt{1}$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.4

Any root of $1$ is $1$ .

$x = \pm 1$

$y = \sqrt{- 7 + 6 x^{2}}$

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.

$x = 1$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.5.2

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

$x = - 1$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.2.5.3

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

$x = 1, - 1$

$y = \sqrt{- 7 + 6 x^{2}}$

$x = 1, - 1$

$y = \sqrt{- 7 + 6 x^{2}}$

$x = 1, - 1$

$y = \sqrt{- 7 + 6 x^{2}}$

Step 2.3

Replace all occurrences of $x$ with $1$ in each equation.

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Step 2.3.1

Replace all occurrences of $x$ in $y = \sqrt{- 7 + 6 x^{2}}$ with $1$ .

$y = \sqrt{- 7 + 6 {(1)}^{2}}$

$x = 1$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{- 7 + 6 {(1)}^{2}}$ .

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Step 2.3.2.1.1

One to any power is one.

$y = \sqrt{- 7 + 6 \cdot 1}$

$x = 1$

Step 2.3.2.1.2

Multiply $6$ by $1$ .

$y = \sqrt{- 7 + 6}$

$x = 1$

Step 2.3.2.1.3

Add $- 7$ and $6$ .

$y = \sqrt{- 1}$

$x = 1$

Step 2.3.2.1.4

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

$y = i$

$x = 1$

$y = i$

$x = 1$

$y = i$

$x = 1$

$y = i$

$x = 1$

Step 2.4

Replace all occurrences of $x$ with $- 1$ in each equation.

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Step 2.4.1

Replace all occurrences of $x$ in $y = \sqrt{- 7 + 6 x^{2}}$ with $- 1$ .

$y = \sqrt{- 7 + 6 {(- 1)}^{2}}$

$x = - 1$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{- 7 + 6 {(- 1)}^{2}}$ .

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Step 2.4.2.1.1

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

$y = \sqrt{- 7 + 6 \cdot 1}$

$x = - 1$

Step 2.4.2.1.2

Multiply $6$ by $1$ .

$y = \sqrt{- 7 + 6}$

$x = - 1$

Step 2.4.2.1.3

Add $- 7$ and $6$ .

$y = \sqrt{- 1}$

$x = - 1$

Step 2.4.2.1.4

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

$y = i$

$x = - 1$

$y = i$

$x = - 1$

$y = i$

$x = - 1$

$y = i$

$x = - 1$

$y = i$

$x = - 1$

Step 3

Solve the system $y = - \sqrt{- 7 + 6 x^{2}}, 7 x^{2} + 2 y^{2} = 5$ .

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Step 3.1

Replace all occurrences of $y$ with $- \sqrt{- 7 + 6 x^{2}}$ in each equation.

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Step 3.1.1

Replace all occurrences of $y$ in $7 x^{2} + 2 y^{2} = 5$ with $- \sqrt{- 7 + 6 x^{2}}$ .

$7 x^{2} + 2 {(- \sqrt{- 7 + 6 x^{2}})}^{2} = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $7 x^{2} + 2 {(- \sqrt{- 7 + 6 x^{2}})}^{2}$ .

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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 $- \sqrt{- 7 + 6 x^{2}}$ .

$7 x^{2} + 2 ({(- 1)}^{2} {\sqrt{- 7 + 6 x^{2}}}^{2}) = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.2

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

$7 x^{2} + 2 (1 {\sqrt{- 7 + 6 x^{2}}}^{2}) = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{- 7 + 6 x^{2}}}^{2}$ by $1$ .

$7 x^{2} + 2 {\sqrt{- 7 + 6 x^{2}}}^{2} = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{- 7 + 6 x^{2}}}^{2}$ as $- 7 + 6 x^{2}$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 7 + 6 x^{2}}$ as ${(- 7 + 6 x^{2})}^{\frac{1}{2}}$ .

$7 x^{2} + 2 {({(- 7 + 6 x^{2})}^{\frac{1}{2}})}^{2} = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.4.2

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

$7 x^{2} + 2 {(- 7 + 6 x^{2})}^{\frac{1}{2} \cdot 2} = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.4.3

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

$7 x^{2} + 2 {(- 7 + 6 x^{2})}^{\frac{2}{2}} = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

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.

$7 x^{2} + 2 {(- 7 + 6 x^{2})}^{\frac{2}{2}} = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$7 x^{2} + 2 (- 7 + 6 x^{2}) = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 (- 7 + 6 x^{2}) = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.4.5

Simplify.

$7 x^{2} + 2 (- 7 + 6 x^{2}) = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} + 2 (- 7 + 6 x^{2}) = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.5

Apply the distributive property.

$7 x^{2} + 2 \cdot - 7 + 2 (6 x^{2}) = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.6

Multiply $2$ by $- 7$ .

$7 x^{2} - 14 + 2 (6 x^{2}) = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.1.7

Multiply $6$ by $2$ .

$7 x^{2} - 14 + 12 x^{2} = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

$7 x^{2} - 14 + 12 x^{2} = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.1.2.1.2

Add $7 x^{2}$ and $12 x^{2}$ .

$19 x^{2} - 14 = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

$19 x^{2} - 14 = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

$19 x^{2} - 14 = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

$19 x^{2} - 14 = 5$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2

Solve for $x$ in $19 x^{2} - 14 = 5$ .

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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

Add $14$ to both sides of the equation.

$19 x^{2} = 5 + 14$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.1.2

Add $5$ and $14$ .

$19 x^{2} = 19$

$y = - \sqrt{- 7 + 6 x^{2}}$

$19 x^{2} = 19$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.2

Divide each term in $19 x^{2} = 19$ by $19$ and simplify.

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Step 3.2.2.1

Divide each term in $19 x^{2} = 19$ by $19$ .

$\frac{19 x^{2}}{19} = \frac{19}{19}$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $19$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{19 x^{2}}{19} = \frac{19}{19}$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.2.2.1.2

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

$x^{2} = \frac{19}{19}$

$y = - \sqrt{- 7 + 6 x^{2}}$

$x^{2} = \frac{19}{19}$

$y = - \sqrt{- 7 + 6 x^{2}}$

$x^{2} = \frac{19}{19}$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $19$ by $19$ .

$x^{2} = 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

$x^{2} = 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

$x^{2} = 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.3

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

$x = \pm \sqrt{1}$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.4

Any root of $1$ is $1$ .

$x = \pm 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

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.

$x = 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.5.2

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

$x = - 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.2.5.3

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

$x = 1, - 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

$x = 1, - 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

$x = 1, - 1$

$y = - \sqrt{- 7 + 6 x^{2}}$

Step 3.3

Replace all occurrences of $x$ with $1$ in each equation.

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Step 3.3.1

Replace all occurrences of $x$ in $y = - \sqrt{- 7 + 6 x^{2}}$ with $1$ .

$y = - \sqrt{- 7 + 6 {(1)}^{2}}$

$x = 1$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{- 7 + 6 {(1)}^{2}}$ .

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Step 3.3.2.1.1

One to any power is one.

$y = - \sqrt{- 7 + 6 \cdot 1}$

$x = 1$

Step 3.3.2.1.2

Multiply $6$ by $1$ .

$y = - \sqrt{- 7 + 6}$

$x = 1$

Step 3.3.2.1.3

Add $- 7$ and $6$ .

$y = - \sqrt{- 1}$

$x = 1$

Step 3.3.2.1.4

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

$y = - i$

$x = 1$

$y = - i$

$x = 1$

$y = - i$

$x = 1$

$y = - i$

$x = 1$

Step 3.4

Replace all occurrences of $x$ with $- 1$ in each equation.

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Step 3.4.1

Replace all occurrences of $x$ in $y = - \sqrt{- 7 + 6 x^{2}}$ with $- 1$ .

$y = - \sqrt{- 7 + 6 {(- 1)}^{2}}$

$x = - 1$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{- 7 + 6 {(- 1)}^{2}}$ .

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Step 3.4.2.1.1

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

$y = - \sqrt{- 7 + 6 \cdot 1}$

$x = - 1$

Step 3.4.2.1.2

Multiply $6$ by $1$ .

$y = - \sqrt{- 7 + 6}$

$x = - 1$

Step 3.4.2.1.3

Add $- 7$ and $6$ .

$y = - \sqrt{- 1}$

$x = - 1$

Step 3.4.2.1.4

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

$y = - i$

$x = - 1$

$y = - i$

$x = - 1$

$y = - i$

$x = - 1$

$y = - i$

$x = - 1$

$y = - i$

$x = - 1$

Step 4

List all of the solutions.

$y = i, x = 1$

$y = i, x = - 1$

$y = - i, x = 1$

$y = - i, x = - 1$

Step 5