Precalculus Examples

$x^{2} + y^{2} = 36$ , $\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

Step 1

Solve for $x$ in $x^{2} + y^{2} = 36$ .

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Step 1.1

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

$x^{2} = 36 - y^{2}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

Step 1.2

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

$x = \pm \sqrt{36 - y^{2}}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

Step 1.3

Simplify $\pm \sqrt{36 - y^{2}}$ .

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Step 1.3.1

Rewrite $36$ as $6^{2}$ .

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

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = y$ .

$x = \pm \sqrt{(6 + y) (6 - y)}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \pm \sqrt{(6 + y) (6 - y)}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

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.

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

Step 1.4.2

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

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

Step 1.4.3

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

$x = \sqrt{(6 + y) (6 - y)}$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$

Step 2

Solve the system $x = \sqrt{(6 + y) (6 - y)}, \frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{(6 + y) (6 - y)}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$ with $\sqrt{(6 + y) (6 - y)}$ .

$\frac{{(\sqrt{(6 + y) (6 - y)})}^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $\frac{{(\sqrt{(6 + y) (6 - y)})}^{2}}{25} + \frac{y^{2}}{49}$ .

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Step 2.1.2.1.1

Simplify the numerator.

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Step 2.1.2.1.1.1

Rewrite ${\sqrt{(6 + y) (6 - y)}}^{2}$ as $(6 + y) (6 - y)$ .

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Step 2.1.2.1.1.1.1

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

$\frac{{({((6 + y) (6 - y))}^{\frac{1}{2}})}^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.1.2

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

$\frac{{((6 + y) (6 - y))}^{\frac{1}{2} \cdot 2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.1.3

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

$\frac{{((6 + y) (6 - y))}^{\frac{2}{2}}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

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.

$\frac{{((6 + y) (6 - y))}^{\frac{2}{2}}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$\frac{(6 + y) (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{(6 + y) (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.1.5

Simplify.

$\frac{(6 + y) (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{(6 + y) (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.2

Expand $(6 + y) (6 - y)$ using the FOIL Method.

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Step 2.1.2.1.1.2.1

Apply the distributive property.

$\frac{6 (6 - y) + y (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$\frac{6 \cdot 6 + 6 (- y) + y (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.2.3

Apply the distributive property.

$\frac{6 \cdot 6 + 6 (- y) + y \cdot 6 + y (- y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{6 \cdot 6 + 6 (- y) + y \cdot 6 + y (- y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.3

Simplify and combine like terms.

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Step 2.1.2.1.1.3.1

Simplify each term.

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Step 2.1.2.1.1.3.1.1

Multiply $6$ by $6$ .

$\frac{36 + 6 (- y) + y \cdot 6 + y (- y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.3.1.2

Multiply $- 1$ by $6$ .

$\frac{36 - 6 y + y \cdot 6 + y (- y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.3.1.3

Move $6$ to the left of $y$ .

$\frac{36 - 6 y + 6 \cdot y + y (- y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{36 - 6 y + 6 y - y \cdot y}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.2.1.1.3.1.5.1

Move $y$ .

$\frac{36 - 6 y + 6 y - (y \cdot y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$\frac{36 - 6 y + 6 y - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{36 - 6 y + 6 y - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{36 - 6 y + 6 y - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.3.2

Add $- 6 y$ and $6 y$ .

$\frac{36 + 0 - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.3.3

Add $36$ and $0$ .

$\frac{36 - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{36 - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.4

Rewrite $36$ as $6^{2}$ .

$\frac{6^{2} - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = y$ .

$\frac{(6 + y) (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{(6 + y) (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.2

To write $\frac{(6 + y) (6 - y)}{25}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$\frac{(6 + y) (6 - y)}{25} \cdot \frac{49}{49} + \frac{y^{2}}{49} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.3

To write $\frac{y^{2}}{49}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{(6 + y) (6 - y)}{25} \cdot \frac{49}{49} + \frac{y^{2}}{49} \cdot \frac{25}{25} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.4

Write each expression with a common denominator of $1225$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.2.1.4.1

Multiply $\frac{(6 + y) (6 - y)}{25}$ by $\frac{49}{49}$ .

$\frac{(6 + y) (6 - y) \cdot 49}{25 \cdot 49} + \frac{y^{2}}{49} \cdot \frac{25}{25} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.4.2

Multiply $25$ by $49$ .

$\frac{(6 + y) (6 - y) \cdot 49}{1225} + \frac{y^{2}}{49} \cdot \frac{25}{25} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.4.3

Multiply $\frac{y^{2}}{49}$ by $\frac{25}{25}$ .

$\frac{(6 + y) (6 - y) \cdot 49}{1225} + \frac{y^{2} \cdot 25}{49 \cdot 25} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.4.4

Multiply $49$ by $25$ .

$\frac{(6 + y) (6 - y) \cdot 49}{1225} + \frac{y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{(6 + y) (6 - y) \cdot 49}{1225} + \frac{y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.5

Combine the numerators over the common denominator.

$\frac{(6 + y) (6 - y) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6

Simplify the numerator.

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Step 2.1.2.1.6.1

Expand $(6 + y) (6 - y)$ using the FOIL Method.

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Step 2.1.2.1.6.1.1

Apply the distributive property.

$\frac{(6 (6 - y) + y (6 - y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.1.2

Apply the distributive property.

$\frac{(6 \cdot 6 + 6 (- y) + y (6 - y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.1.3

Apply the distributive property.

$\frac{(6 \cdot 6 + 6 (- y) + y \cdot 6 + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{(6 \cdot 6 + 6 (- y) + y \cdot 6 + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.2

Simplify and combine like terms.

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Step 2.1.2.1.6.2.1

Simplify each term.

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Step 2.1.2.1.6.2.1.1

Multiply $6$ by $6$ .

$\frac{(36 + 6 (- y) + y \cdot 6 + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.2.1.2

Multiply $- 1$ by $6$ .

$\frac{(36 - 6 y + y \cdot 6 + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.2.1.3

Move $6$ to the left of $y$ .

$\frac{(36 - 6 y + 6 \cdot y + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{(36 - 6 y + 6 y - y \cdot y) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.2.1.6.2.1.5.1

Move $y$ .

$\frac{(36 - 6 y + 6 y - (y \cdot y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.2.1.5.2

Multiply $y$ by $y$ .

$\frac{(36 - 6 y + 6 y - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{(36 - 6 y + 6 y - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{(36 - 6 y + 6 y - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.2.2

Add $- 6 y$ and $6 y$ .

$\frac{(36 + 0 - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.2.3

Add $36$ and $0$ .

$\frac{(36 - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{(36 - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.3

Apply the distributive property.

$\frac{36 \cdot 49 - y^{2} \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.4

Multiply $36$ by $49$ .

$\frac{1764 - y^{2} \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.5

Multiply $49$ by $- 1$ .

$\frac{1764 - 49 y^{2} + y^{2} \cdot 25}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.6

Move $25$ to the left of $y^{2}$ .

$\frac{1764 - 49 y^{2} + 25 \cdot y^{2}}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.7

Add $- 49 y^{2}$ and $25 y^{2}$ .

$\frac{1764 - 24 y^{2}}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.8

Factor $12$ out of $1764 - 24 y^{2}$ .

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Step 2.1.2.1.6.8.1

Factor $12$ out of $1764$ .

$\frac{12 (147) - 24 y^{2}}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.8.2

Factor $12$ out of $- 24 y^{2}$ .

$\frac{12 (147) + 12 (- 2 y^{2})}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.1.2.1.6.8.3

Factor $12$ out of $12 (147) + 12 (- 2 y^{2})$ .

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2

Solve for $y$ in $\frac{12 (147 - 2 y^{2})}{1225} = 1$ .

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Step 2.2.1

Multiply both sides by $1225$ .

$\frac{12 (147 - 2 y^{2})}{1225} \cdot 1225 = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify the left side.

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Step 2.2.2.1.1

Simplify $\frac{12 (147 - 2 y^{2})}{1225} \cdot 1225$ .

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Step 2.2.2.1.1.1

Cancel the common factor of $1225$ .

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Step 2.2.2.1.1.1.1

Cancel the common factor.

$\frac{12 (147 - 2 y^{2})}{1225} \cdot 1225 = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.2.1.1.1.2

Rewrite the expression.

$12 (147 - 2 y^{2}) = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

$12 (147 - 2 y^{2}) = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.2.1.1.2

Apply the distributive property.

$12 \cdot 147 + 12 (- 2 y^{2}) = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.2.1.1.3

Simplify the expression.

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Step 2.2.2.1.1.3.1

Multiply $12$ by $147$ .

$1764 + 12 (- 2 y^{2}) = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.2.1.1.3.2

Multiply $- 2$ by $12$ .

$1764 - 24 y^{2} = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.2.1.1.3.3

Reorder $1764$ and $- 24 y^{2}$ .

$- 24 y^{2} + 1764 = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1 \cdot 1225$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.2.2

Simplify the right side.

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Step 2.2.2.2.1

Multiply $1225$ by $1$ .

$- 24 y^{2} + 1764 = 1225$

$x = \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1225$

$x = \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1225$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3

Solve for $y$ .

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Step 2.2.3.1

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

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Step 2.2.3.1.1

Subtract $1764$ from both sides of the equation.

$- 24 y^{2} = 1225 - 1764$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.1.2

Subtract $1764$ from $1225$ .

$- 24 y^{2} = - 539$

$x = \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} = - 539$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.2

Divide each term in $- 24 y^{2} = - 539$ by $- 24$ and simplify.

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Step 2.2.3.2.1

Divide each term in $- 24 y^{2} = - 539$ by $- 24$ .

$\frac{- 24 y^{2}}{- 24} = \frac{- 539}{- 24}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.2.2

Simplify the left side.

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Step 2.2.3.2.2.1

Cancel the common factor of $- 24$ .

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Step 2.2.3.2.2.1.1

Cancel the common factor.

$\frac{- 24 y^{2}}{- 24} = \frac{- 539}{- 24}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.2.2.1.2

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

$y^{2} = \frac{- 539}{- 24}$

$x = \sqrt{(6 + y) (6 - y)}$

$y^{2} = \frac{- 539}{- 24}$

$x = \sqrt{(6 + y) (6 - y)}$

$y^{2} = \frac{- 539}{- 24}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.2.3

Simplify the right side.

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Step 2.2.3.2.3.1

Dividing two negative values results in a positive value.

$y^{2} = \frac{539}{24}$

$x = \sqrt{(6 + y) (6 - y)}$

$y^{2} = \frac{539}{24}$

$x = \sqrt{(6 + y) (6 - y)}$

$y^{2} = \frac{539}{24}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.3

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

$y = \pm \sqrt{\frac{539}{24}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4

Simplify $\pm \sqrt{\frac{539}{24}}$ .

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Step 2.2.3.4.1

Rewrite $\sqrt{\frac{539}{24}}$ as $\frac{\sqrt{539}}{\sqrt{24}}$ .

$y = \pm \frac{\sqrt{539}}{\sqrt{24}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.2

Simplify the numerator.

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Step 2.2.3.4.2.1

Rewrite $539$ as $7^{2} \cdot 11$ .

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Step 2.2.3.4.2.1.1

Factor $49$ out of $539$ .

$y = \pm \frac{\sqrt{49 (11)}}{\sqrt{24}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.2.1.2

Rewrite $49$ as $7^{2}$ .

$y = \pm \frac{\sqrt{7^{2} \cdot 11}}{\sqrt{24}}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{\sqrt{7^{2} \cdot 11}}{\sqrt{24}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.2.2

Pull terms out from under the radical.

$y = \pm \frac{7 \sqrt{11}}{\sqrt{24}}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11}}{\sqrt{24}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.3

Simplify the denominator.

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Step 2.2.3.4.3.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.2.3.4.3.1.1

Factor $4$ out of $24$ .

$y = \pm \frac{7 \sqrt{11}}{\sqrt{4 (6)}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.3.1.2

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

$y = \pm \frac{7 \sqrt{11}}{\sqrt{2^{2} \cdot 6}}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11}}{\sqrt{2^{2} \cdot 6}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.3.2

Pull terms out from under the radical.

$y = \pm \frac{7 \sqrt{11}}{2 \sqrt{6}}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11}}{2 \sqrt{6}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.4

Multiply $\frac{7 \sqrt{11}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{7 \sqrt{11}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5

Combine and simplify the denominator.

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Step 2.2.3.4.5.1

Multiply $\frac{7 \sqrt{11}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.2

Move $\sqrt{6}$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.4

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.5

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

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.6

Add $1$ and $1$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 {\sqrt{6}}^{2}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.2.3.4.5.7.1

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

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.7.2

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

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.7.3

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

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.7.4

Cancel the common factor of $2$ .

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Step 2.2.3.4.5.7.4.1

Cancel the common factor.

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.7.4.2

Rewrite the expression.

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.5.7.5

Evaluate the exponent.

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.6

Simplify the numerator.

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Step 2.2.3.4.6.1

Combine using the product rule for radicals.

$y = \pm \frac{7 \sqrt{6 \cdot 11}}{2 \cdot 6}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.6.2

Multiply $6$ by $11$ .

$y = \pm \frac{7 \sqrt{66}}{2 \cdot 6}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{66}}{2 \cdot 6}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.4.7

Multiply $2$ by $6$ .

$y = \pm \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.5

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

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Step 2.2.3.5.1

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

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.5.2

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

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.2.3.5.3

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

$y = \frac{7 \sqrt{66}}{12}, - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \frac{7 \sqrt{66}}{12}, - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \frac{7 \sqrt{66}}{12}, - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + y) (6 - y)}$

$y = \frac{7 \sqrt{66}}{12}, - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + y) (6 - y)}$

Step 2.3

Replace all occurrences of $y$ with $\frac{7 \sqrt{66}}{12}$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{(6 + y) (6 - y)}$ with $\frac{7 \sqrt{66}}{12}$ .

$x = \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2

Simplify $x = \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$ .

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Step 2.3.2.1

Simplify the left side.

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Step 2.3.2.1.1

Remove parentheses.

$x = \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $\sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$ .

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Step 2.3.2.2.1.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = \sqrt{(6 \cdot \frac{12}{12} + \frac{7 \sqrt{66}}{12}) (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.2

Combine $6$ and $\frac{12}{12}$ .

$x = \sqrt{(\frac{6 \cdot 12}{12} + \frac{7 \sqrt{66}}{12}) (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.3

Simplify the expression.

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Step 2.3.2.2.1.3.1

Combine the numerators over the common denominator.

$x = \sqrt{\frac{6 \cdot 12 + 7 \sqrt{66}}{12} \cdot (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.3.2

Multiply $6$ by $12$ .

$x = \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.4

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot (6 \cdot \frac{12}{12} - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.5

Combine fractions.

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Step 2.3.2.2.1.5.1

Combine $6$ and $\frac{12}{12}$ .

$x = \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot (\frac{6 \cdot 12}{12} - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.5.2

Simplify the expression.

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Step 2.3.2.2.1.5.2.1

Combine the numerators over the common denominator.

$x = \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot \frac{6 \cdot 12 - 7 \sqrt{66}}{12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.5.2.2

Multiply $6$ by $12$ .

$x = \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot \frac{72 - 7 \sqrt{66}}{12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot \frac{72 - 7 \sqrt{66}}{12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.5.3

Multiply $\frac{72 + 7 \sqrt{66}}{12}$ by $\frac{72 - 7 \sqrt{66}}{12}$ .

$x = \sqrt{\frac{(72 + 7 \sqrt{66}) (72 - 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{(72 + 7 \sqrt{66}) (72 - 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6

Simplify the numerator.

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Step 2.3.2.2.1.6.1

Expand $(72 + 7 \sqrt{66}) (72 - 7 \sqrt{66})$ using the FOIL Method.

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Step 2.3.2.2.1.6.1.1

Apply the distributive property.

$x = \sqrt{\frac{72 (72 - 7 \sqrt{66}) + 7 \sqrt{66} (72 - 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.1.2

Apply the distributive property.

$x = \sqrt{\frac{72 \cdot 72 + 72 (- 7 \sqrt{66}) + 7 \sqrt{66} (72 - 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.1.3

Apply the distributive property.

$x = \sqrt{\frac{72 \cdot 72 + 72 (- 7 \sqrt{66}) + 7 \sqrt{66} \cdot 72 + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{72 \cdot 72 + 72 (- 7 \sqrt{66}) + 7 \sqrt{66} \cdot 72 + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2

Simplify and combine like terms.

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Step 2.3.2.2.1.6.2.1

Simplify each term.

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Step 2.3.2.2.1.6.2.1.1

Multiply $72$ by $72$ .

$x = \sqrt{\frac{5184 + 72 (- 7 \sqrt{66}) + 7 \sqrt{66} \cdot 72 + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.2

Multiply $- 7$ by $72$ .

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 7 \sqrt{66} \cdot 72 + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.3

Multiply $72$ by $7$ .

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.4

Multiply $7 \sqrt{66} (- 7 \sqrt{66})$ .

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Step 2.3.2.2.1.6.2.1.4.1

Multiply $- 7$ by $7$ .

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \sqrt{66} \sqrt{66}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.4.2

Raise $\sqrt{66}$ to the power of $1$ .

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 (\sqrt{66} \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.4.3

Raise $\sqrt{66}$ to the power of $1$ .

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 (\sqrt{66} \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.4.4

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

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 {\sqrt{66}}^{1 + 1}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.4.5

Add $1$ and $1$ .

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 {\sqrt{66}}^{2}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 {\sqrt{66}}^{2}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.5

Rewrite ${\sqrt{66}}^{2}$ as $66$ .

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Step 2.3.2.2.1.6.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{66}$ as $66^{\frac{1}{2}}$ .

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 {(66^{\frac{1}{2}})}^{2}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.5.2

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

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66^{\frac{1}{2} \cdot 2}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.5.3

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

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66^{\frac{2}{2}}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.6.2.1.5.4.1

Cancel the common factor.

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66^{\frac{2}{2}}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.5.4.2

Rewrite the expression.

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.5.5

Evaluate the exponent.

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.1.6

Multiply $- 49$ by $66$ .

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 3234}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 3234}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.2

Subtract $3234$ from $5184$ .

$x = \sqrt{\frac{1950 - 504 \sqrt{66} + 504 \sqrt{66}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.3

Add $- 504 \sqrt{66}$ and $504 \sqrt{66}$ .

$x = \sqrt{\frac{1950 + 0}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.6.2.4

Add $1950$ and $0$ .

$x = \sqrt{\frac{1950}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{1950}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{1950}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.7

Reduce the expression by cancelling the common factors.

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Step 2.3.2.2.1.7.1

Multiply $12$ by $12$ .

$x = \sqrt{\frac{1950}{144}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.7.2

Cancel the common factor of $1950$ and $144$ .

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Step 2.3.2.2.1.7.2.1

Factor $6$ out of $1950$ .

$x = \sqrt{\frac{6 (325)}{144}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.7.2.2

Cancel the common factors.

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Step 2.3.2.2.1.7.2.2.1

Factor $6$ out of $144$ .

$x = \sqrt{\frac{6 \cdot 325}{6 \cdot 24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.7.2.2.2

Cancel the common factor.

$x = \sqrt{\frac{6 \cdot 325}{6 \cdot 24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.7.2.2.3

Rewrite the expression.

$x = \sqrt{\frac{325}{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{325}{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{325}{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{325}{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.8

Rewrite $\sqrt{\frac{325}{24}}$ as $\frac{\sqrt{325}}{\sqrt{24}}$ .

$x = \frac{\sqrt{325}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.9

Simplify the numerator.

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Step 2.3.2.2.1.9.1

Rewrite $325$ as $5^{2} \cdot 13$ .

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Step 2.3.2.2.1.9.1.1

Factor $25$ out of $325$ .

$x = \frac{\sqrt{25 (13)}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.9.1.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{\sqrt{5^{2} \cdot 13}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{\sqrt{5^{2} \cdot 13}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.9.2

Pull terms out from under the radical.

$x = \frac{5 \sqrt{13}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.10

Simplify the denominator.

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Step 2.3.2.2.1.10.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.3.2.2.1.10.1.1

Factor $4$ out of $24$ .

$x = \frac{5 \sqrt{13}}{\sqrt{4 (6)}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.10.1.2

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

$x = \frac{5 \sqrt{13}}{\sqrt{2^{2} \cdot 6}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13}}{\sqrt{2^{2} \cdot 6}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.10.2

Pull terms out from under the radical.

$x = \frac{5 \sqrt{13}}{2 \sqrt{6}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13}}{2 \sqrt{6}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.11

Multiply $\frac{5 \sqrt{13}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \frac{5 \sqrt{13}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12

Combine and simplify the denominator.

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Step 2.3.2.2.1.12.1

Multiply $\frac{5 \sqrt{13}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.2

Move $\sqrt{6}$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.3

Raise $\sqrt{6}$ to the power of $1$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.4

Raise $\sqrt{6}$ to the power of $1$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.5

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

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.6

Add $1$ and $1$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 {\sqrt{6}}^{2}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.3.2.2.1.12.7.1

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

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.7.2

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

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.7.3

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

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.7.4

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.12.7.4.1

Cancel the common factor.

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.7.4.2

Rewrite the expression.

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.12.7.5

Evaluate the exponent.

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.13

Simplify the numerator.

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Step 2.3.2.2.1.13.1

Combine using the product rule for radicals.

$x = \frac{5 \sqrt{6 \cdot 13}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.13.2

Multiply $6$ by $13$ .

$x = \frac{5 \sqrt{78}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.3.2.2.1.14

Multiply $2$ by $6$ .

$x = \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

Step 2.4

Replace all occurrences of $y$ with $- \frac{7 \sqrt{66}}{12}$ in each equation.

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Step 2.4.1

Replace all occurrences of $y$ in $x = \sqrt{(6 + y) (6 - y)}$ with $- \frac{7 \sqrt{66}}{12}$ .

$x = \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2

Simplify $x = \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$ .

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Step 2.4.2.1

Simplify the left side.

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Step 2.4.2.1.1

Remove parentheses.

$x = \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2

Simplify the right side.

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Step 2.4.2.2.1

Simplify $\sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$ .

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Step 2.4.2.2.1.1

Multiply $- - \frac{7 \sqrt{66}}{12}$ .

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Step 2.4.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$x = \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 + 1 (\frac{7 \sqrt{66}}{12}))}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.1.2

Multiply $\frac{7 \sqrt{66}}{12}$ by $1$ .

$x = \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = \sqrt{(6 \cdot \frac{12}{12} - \frac{7 \sqrt{66}}{12}) (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.3

Combine $6$ and $\frac{12}{12}$ .

$x = \sqrt{(\frac{6 \cdot 12}{12} - \frac{7 \sqrt{66}}{12}) (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.4

Simplify the expression.

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Step 2.4.2.2.1.4.1

Combine the numerators over the common denominator.

$x = \sqrt{\frac{6 \cdot 12 - 7 \sqrt{66}}{12} \cdot (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.4.2

Multiply $6$ by $12$ .

$x = \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.5

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot (6 \cdot \frac{12}{12} + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.6

Combine fractions.

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Step 2.4.2.2.1.6.1

Combine $6$ and $\frac{12}{12}$ .

$x = \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot (\frac{6 \cdot 12}{12} + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.6.2

Simplify the expression.

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Step 2.4.2.2.1.6.2.1

Combine the numerators over the common denominator.

$x = \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot \frac{6 \cdot 12 + 7 \sqrt{66}}{12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.6.2.2

Multiply $6$ by $12$ .

$x = \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot \frac{72 + 7 \sqrt{66}}{12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot \frac{72 + 7 \sqrt{66}}{12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.6.3

Multiply $\frac{72 - 7 \sqrt{66}}{12}$ by $\frac{72 + 7 \sqrt{66}}{12}$ .

$x = \sqrt{\frac{(72 - 7 \sqrt{66}) (72 + 7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{(72 - 7 \sqrt{66}) (72 + 7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7

Simplify the numerator.

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Step 2.4.2.2.1.7.1

Expand $(72 - 7 \sqrt{66}) (72 + 7 \sqrt{66})$ using the FOIL Method.

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Step 2.4.2.2.1.7.1.1

Apply the distributive property.

$x = \sqrt{\frac{72 (72 + 7 \sqrt{66}) - 7 \sqrt{66} (72 + 7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.1.2

Apply the distributive property.

$x = \sqrt{\frac{72 \cdot 72 + 72 (7 \sqrt{66}) - 7 \sqrt{66} (72 + 7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.1.3

Apply the distributive property.

$x = \sqrt{\frac{72 \cdot 72 + 72 (7 \sqrt{66}) - 7 \sqrt{66} \cdot 72 - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{72 \cdot 72 + 72 (7 \sqrt{66}) - 7 \sqrt{66} \cdot 72 - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2

Simplify and combine like terms.

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Step 2.4.2.2.1.7.2.1

Simplify each term.

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Step 2.4.2.2.1.7.2.1.1

Multiply $72$ by $72$ .

$x = \sqrt{\frac{5184 + 72 (7 \sqrt{66}) - 7 \sqrt{66} \cdot 72 - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.2

Multiply $7$ by $72$ .

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 7 \sqrt{66} \cdot 72 - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.3

Multiply $72$ by $- 7$ .

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.4

Multiply $- 7 \sqrt{66} (7 \sqrt{66})$ .

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Step 2.4.2.2.1.7.2.1.4.1

Multiply $7$ by $- 7$ .

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \sqrt{66} \sqrt{66}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.4.2

Raise $\sqrt{66}$ to the power of $1$ .

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 (\sqrt{66} \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.4.3

Raise $\sqrt{66}$ to the power of $1$ .

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 (\sqrt{66} \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.4.4

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

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 {\sqrt{66}}^{1 + 1}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.4.5

Add $1$ and $1$ .

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 {\sqrt{66}}^{2}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 {\sqrt{66}}^{2}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.5

Rewrite ${\sqrt{66}}^{2}$ as $66$ .

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Step 2.4.2.2.1.7.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{66}$ as $66^{\frac{1}{2}}$ .

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 {(66^{\frac{1}{2}})}^{2}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.5.2

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

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66^{\frac{1}{2} \cdot 2}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.5.3

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

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66^{\frac{2}{2}}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.5.4

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.7.2.1.5.4.1

Cancel the common factor.

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66^{\frac{2}{2}}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.5.4.2

Rewrite the expression.

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.5.5

Evaluate the exponent.

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.1.6

Multiply $- 49$ by $66$ .

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 3234}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 3234}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.2

Subtract $3234$ from $5184$ .

$x = \sqrt{\frac{1950 + 504 \sqrt{66} - 504 \sqrt{66}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.3

Subtract $504 \sqrt{66}$ from $504 \sqrt{66}$ .

$x = \sqrt{\frac{1950 + 0}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.7.2.4

Add $1950$ and $0$ .

$x = \sqrt{\frac{1950}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{1950}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{1950}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.8

Reduce the expression by cancelling the common factors.

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Step 2.4.2.2.1.8.1

Multiply $12$ by $12$ .

$x = \sqrt{\frac{1950}{144}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.8.2

Cancel the common factor of $1950$ and $144$ .

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Step 2.4.2.2.1.8.2.1

Factor $6$ out of $1950$ .

$x = \sqrt{\frac{6 (325)}{144}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.8.2.2

Cancel the common factors.

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Step 2.4.2.2.1.8.2.2.1

Factor $6$ out of $144$ .

$x = \sqrt{\frac{6 \cdot 325}{6 \cdot 24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.8.2.2.2

Cancel the common factor.

$x = \sqrt{\frac{6 \cdot 325}{6 \cdot 24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.8.2.2.3

Rewrite the expression.

$x = \sqrt{\frac{325}{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{325}{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{325}{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \sqrt{\frac{325}{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.9

Rewrite $\sqrt{\frac{325}{24}}$ as $\frac{\sqrt{325}}{\sqrt{24}}$ .

$x = \frac{\sqrt{325}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.10

Simplify the numerator.

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Step 2.4.2.2.1.10.1

Rewrite $325$ as $5^{2} \cdot 13$ .

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Step 2.4.2.2.1.10.1.1

Factor $25$ out of $325$ .

$x = \frac{\sqrt{25 (13)}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.10.1.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{\sqrt{5^{2} \cdot 13}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{\sqrt{5^{2} \cdot 13}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.10.2

Pull terms out from under the radical.

$x = \frac{5 \sqrt{13}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.11

Simplify the denominator.

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Step 2.4.2.2.1.11.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.4.2.2.1.11.1.1

Factor $4$ out of $24$ .

$x = \frac{5 \sqrt{13}}{\sqrt{4 (6)}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.11.1.2

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

$x = \frac{5 \sqrt{13}}{\sqrt{2^{2} \cdot 6}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13}}{\sqrt{2^{2} \cdot 6}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.11.2

Pull terms out from under the radical.

$x = \frac{5 \sqrt{13}}{2 \sqrt{6}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13}}{2 \sqrt{6}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.12

Multiply $\frac{5 \sqrt{13}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \frac{5 \sqrt{13}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13

Combine and simplify the denominator.

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Step 2.4.2.2.1.13.1

Multiply $\frac{5 \sqrt{13}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.2

Move $\sqrt{6}$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.3

Raise $\sqrt{6}$ to the power of $1$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.4

Raise $\sqrt{6}$ to the power of $1$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.5

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

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.6

Add $1$ and $1$ .

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 {\sqrt{6}}^{2}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.4.2.2.1.13.7.1

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

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.7.2

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

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.7.3

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

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.7.4

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.13.7.4.1

Cancel the common factor.

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.7.4.2

Rewrite the expression.

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.13.7.5

Evaluate the exponent.

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.14

Simplify the numerator.

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Step 2.4.2.2.1.14.1

Combine using the product rule for radicals.

$x = \frac{5 \sqrt{6 \cdot 13}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.14.2

Multiply $6$ by $13$ .

$x = \frac{5 \sqrt{78}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 2.4.2.2.1.15

Multiply $2$ by $6$ .

$x = \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3

Solve the system $x = - \sqrt{(6 + y) (6 - y)}, \frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{(6 + y) (6 - y)}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $\frac{x^{2}}{25} + \frac{y^{2}}{49} = 1$ with $- \sqrt{(6 + y) (6 - y)}$ .

$\frac{{(- \sqrt{(6 + y) (6 - y)})}^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $\frac{{(- \sqrt{(6 + y) (6 - y)})}^{2}}{25} + \frac{y^{2}}{49}$ .

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Step 3.1.2.1.1

Simplify the numerator.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{(6 + y) (6 - y)}$ .

$\frac{{(- 1)}^{2} {\sqrt{(6 + y) (6 - y)}}^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.2

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

$\frac{1 {\sqrt{(6 + y) (6 - y)}}^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.3

Rewrite ${\sqrt{(6 + y) (6 - y)}}^{2}$ as $(6 + y) (6 - y)$ .

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Step 3.1.2.1.1.3.1

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

$\frac{1 {({((6 + y) (6 - y))}^{\frac{1}{2}})}^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.3.2

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

$\frac{1 {((6 + y) (6 - y))}^{\frac{1}{2} \cdot 2}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.3.3

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

$\frac{1 {((6 + y) (6 - y))}^{\frac{2}{2}}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.3.4.1

Cancel the common factor.

$\frac{1 {((6 + y) (6 - y))}^{\frac{2}{2}}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.3.4.2

Rewrite the expression.

$\frac{1 ((6 + y) (6 - y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{1 ((6 + y) (6 - y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.3.5

Simplify.

$\frac{1 ((6 + y) (6 - y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{1 ((6 + y) (6 - y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.4

Expand $(6 + y) (6 - y)$ using the FOIL Method.

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Step 3.1.2.1.1.4.1

Apply the distributive property.

$\frac{1 (6 (6 - y) + y (6 - y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.4.2

Apply the distributive property.

$\frac{1 (6 \cdot 6 + 6 (- y) + y (6 - y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.4.3

Apply the distributive property.

$\frac{1 (6 \cdot 6 + 6 (- y) + y \cdot 6 + y (- y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{1 (6 \cdot 6 + 6 (- y) + y \cdot 6 + y (- y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.5

Simplify and combine like terms.

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Step 3.1.2.1.1.5.1

Simplify each term.

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Step 3.1.2.1.1.5.1.1

Multiply $6$ by $6$ .

$\frac{1 (36 + 6 (- y) + y \cdot 6 + y (- y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.5.1.2

Multiply $- 1$ by $6$ .

$\frac{1 (36 - 6 y + y \cdot 6 + y (- y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.5.1.3

Move $6$ to the left of $y$ .

$\frac{1 (36 - 6 y + 6 \cdot y + y (- y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.5.1.4

Rewrite using the commutative property of multiplication.

$\frac{1 (36 - 6 y + 6 y - y \cdot y)}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.5.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.1.1.5.1.5.1

Move $y$ .

$\frac{1 (36 - 6 y + 6 y - (y \cdot y))}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.5.1.5.2

Multiply $y$ by $y$ .

$\frac{1 (36 - 6 y + 6 y - y^{2})}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{1 (36 - 6 y + 6 y - y^{2})}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{1 (36 - 6 y + 6 y - y^{2})}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.5.2

Add $- 6 y$ and $6 y$ .

$\frac{1 (36 + 0 - y^{2})}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.5.3

Add $36$ and $0$ .

$\frac{1 (36 - y^{2})}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{1 (36 - y^{2})}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.6

Multiply $36 - y^{2}$ by $1$ .

$\frac{36 - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.7

Rewrite $36$ as $6^{2}$ .

$\frac{6^{2} - y^{2}}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.1.8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = y$ .

$\frac{(6 + y) (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{(6 + y) (6 - y)}{25} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.2

To write $\frac{(6 + y) (6 - y)}{25}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$\frac{(6 + y) (6 - y)}{25} \cdot \frac{49}{49} + \frac{y^{2}}{49} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.3

To write $\frac{y^{2}}{49}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{(6 + y) (6 - y)}{25} \cdot \frac{49}{49} + \frac{y^{2}}{49} \cdot \frac{25}{25} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.4

Write each expression with a common denominator of $1225$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.2.1.4.1

Multiply $\frac{(6 + y) (6 - y)}{25}$ by $\frac{49}{49}$ .

$\frac{(6 + y) (6 - y) \cdot 49}{25 \cdot 49} + \frac{y^{2}}{49} \cdot \frac{25}{25} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.4.2

Multiply $25$ by $49$ .

$\frac{(6 + y) (6 - y) \cdot 49}{1225} + \frac{y^{2}}{49} \cdot \frac{25}{25} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.4.3

Multiply $\frac{y^{2}}{49}$ by $\frac{25}{25}$ .

$\frac{(6 + y) (6 - y) \cdot 49}{1225} + \frac{y^{2} \cdot 25}{49 \cdot 25} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.4.4

Multiply $49$ by $25$ .

$\frac{(6 + y) (6 - y) \cdot 49}{1225} + \frac{y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{(6 + y) (6 - y) \cdot 49}{1225} + \frac{y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.5

Combine the numerators over the common denominator.

$\frac{(6 + y) (6 - y) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6

Simplify the numerator.

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Step 3.1.2.1.6.1

Expand $(6 + y) (6 - y)$ using the FOIL Method.

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Step 3.1.2.1.6.1.1

Apply the distributive property.

$\frac{(6 (6 - y) + y (6 - y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.1.2

Apply the distributive property.

$\frac{(6 \cdot 6 + 6 (- y) + y (6 - y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.1.3

Apply the distributive property.

$\frac{(6 \cdot 6 + 6 (- y) + y \cdot 6 + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{(6 \cdot 6 + 6 (- y) + y \cdot 6 + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.2

Simplify and combine like terms.

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Step 3.1.2.1.6.2.1

Simplify each term.

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Step 3.1.2.1.6.2.1.1

Multiply $6$ by $6$ .

$\frac{(36 + 6 (- y) + y \cdot 6 + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.2.1.2

Multiply $- 1$ by $6$ .

$\frac{(36 - 6 y + y \cdot 6 + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.2.1.3

Move $6$ to the left of $y$ .

$\frac{(36 - 6 y + 6 \cdot y + y (- y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{(36 - 6 y + 6 y - y \cdot y) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.1.6.2.1.5.1

Move $y$ .

$\frac{(36 - 6 y + 6 y - (y \cdot y)) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.2.1.5.2

Multiply $y$ by $y$ .

$\frac{(36 - 6 y + 6 y - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{(36 - 6 y + 6 y - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{(36 - 6 y + 6 y - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.2.2

Add $- 6 y$ and $6 y$ .

$\frac{(36 + 0 - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.2.3

Add $36$ and $0$ .

$\frac{(36 - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{(36 - y^{2}) \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.3

Apply the distributive property.

$\frac{36 \cdot 49 - y^{2} \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.4

Multiply $36$ by $49$ .

$\frac{1764 - y^{2} \cdot 49 + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.5

Multiply $49$ by $- 1$ .

$\frac{1764 - 49 y^{2} + y^{2} \cdot 25}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.6

Move $25$ to the left of $y^{2}$ .

$\frac{1764 - 49 y^{2} + 25 \cdot y^{2}}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.7

Add $- 49 y^{2}$ and $25 y^{2}$ .

$\frac{1764 - 24 y^{2}}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.8

Factor $12$ out of $1764 - 24 y^{2}$ .

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Step 3.1.2.1.6.8.1

Factor $12$ out of $1764$ .

$\frac{12 (147) - 24 y^{2}}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.8.2

Factor $12$ out of $- 24 y^{2}$ .

$\frac{12 (147) + 12 (- 2 y^{2})}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.1.2.1.6.8.3

Factor $12$ out of $12 (147) + 12 (- 2 y^{2})$ .

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

$\frac{12 (147 - 2 y^{2})}{1225} = 1$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2

Solve for $y$ in $\frac{12 (147 - 2 y^{2})}{1225} = 1$ .

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Step 3.2.1

Multiply both sides by $1225$ .

$\frac{12 (147 - 2 y^{2})}{1225} \cdot 1225 = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Simplify $\frac{12 (147 - 2 y^{2})}{1225} \cdot 1225$ .

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Step 3.2.2.1.1.1

Cancel the common factor of $1225$ .

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Step 3.2.2.1.1.1.1

Cancel the common factor.

$\frac{12 (147 - 2 y^{2})}{1225} \cdot 1225 = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.2.1.1.1.2

Rewrite the expression.

$12 (147 - 2 y^{2}) = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

$12 (147 - 2 y^{2}) = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.2.1.1.2

Apply the distributive property.

$12 \cdot 147 + 12 (- 2 y^{2}) = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.2.1.1.3

Simplify the expression.

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Step 3.2.2.1.1.3.1

Multiply $12$ by $147$ .

$1764 + 12 (- 2 y^{2}) = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.2.1.1.3.2

Multiply $- 2$ by $12$ .

$1764 - 24 y^{2} = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.2.1.1.3.3

Reorder $1764$ and $- 24 y^{2}$ .

$- 24 y^{2} + 1764 = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1 \cdot 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Multiply $1225$ by $1$ .

$- 24 y^{2} + 1764 = 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} + 1764 = 1225$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3

Solve for $y$ .

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Step 3.2.3.1

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

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Step 3.2.3.1.1

Subtract $1764$ from both sides of the equation.

$- 24 y^{2} = 1225 - 1764$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.1.2

Subtract $1764$ from $1225$ .

$- 24 y^{2} = - 539$

$x = - \sqrt{(6 + y) (6 - y)}$

$- 24 y^{2} = - 539$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.2

Divide each term in $- 24 y^{2} = - 539$ by $- 24$ and simplify.

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Step 3.2.3.2.1

Divide each term in $- 24 y^{2} = - 539$ by $- 24$ .

$\frac{- 24 y^{2}}{- 24} = \frac{- 539}{- 24}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.2.2

Simplify the left side.

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Step 3.2.3.2.2.1

Cancel the common factor of $- 24$ .

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Step 3.2.3.2.2.1.1

Cancel the common factor.

$\frac{- 24 y^{2}}{- 24} = \frac{- 539}{- 24}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.2.2.1.2

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

$y^{2} = \frac{- 539}{- 24}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y^{2} = \frac{- 539}{- 24}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y^{2} = \frac{- 539}{- 24}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.2.3

Simplify the right side.

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Step 3.2.3.2.3.1

Dividing two negative values results in a positive value.

$y^{2} = \frac{539}{24}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y^{2} = \frac{539}{24}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y^{2} = \frac{539}{24}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.3

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

$y = \pm \sqrt{\frac{539}{24}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4

Simplify $\pm \sqrt{\frac{539}{24}}$ .

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Step 3.2.3.4.1

Rewrite $\sqrt{\frac{539}{24}}$ as $\frac{\sqrt{539}}{\sqrt{24}}$ .

$y = \pm \frac{\sqrt{539}}{\sqrt{24}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.2

Simplify the numerator.

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Step 3.2.3.4.2.1

Rewrite $539$ as $7^{2} \cdot 11$ .

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Step 3.2.3.4.2.1.1

Factor $49$ out of $539$ .

$y = \pm \frac{\sqrt{49 (11)}}{\sqrt{24}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.2.1.2

Rewrite $49$ as $7^{2}$ .

$y = \pm \frac{\sqrt{7^{2} \cdot 11}}{\sqrt{24}}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{\sqrt{7^{2} \cdot 11}}{\sqrt{24}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.2.2

Pull terms out from under the radical.

$y = \pm \frac{7 \sqrt{11}}{\sqrt{24}}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11}}{\sqrt{24}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.3

Simplify the denominator.

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Step 3.2.3.4.3.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 3.2.3.4.3.1.1

Factor $4$ out of $24$ .

$y = \pm \frac{7 \sqrt{11}}{\sqrt{4 (6)}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.3.1.2

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

$y = \pm \frac{7 \sqrt{11}}{\sqrt{2^{2} \cdot 6}}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11}}{\sqrt{2^{2} \cdot 6}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.3.2

Pull terms out from under the radical.

$y = \pm \frac{7 \sqrt{11}}{2 \sqrt{6}}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11}}{2 \sqrt{6}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.4

Multiply $\frac{7 \sqrt{11}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{7 \sqrt{11}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5

Combine and simplify the denominator.

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Step 3.2.3.4.5.1

Multiply $\frac{7 \sqrt{11}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.2

Move $\sqrt{6}$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.4

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.5

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

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.6

Add $1$ and $1$ .

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 {\sqrt{6}}^{2}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.2.3.4.5.7.1

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

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.7.2

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

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.7.3

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

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.7.4

Cancel the common factor of $2$ .

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Step 3.2.3.4.5.7.4.1

Cancel the common factor.

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.7.4.2

Rewrite the expression.

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.5.7.5

Evaluate the exponent.

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{11} \sqrt{6}}{2 \cdot 6}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.6

Simplify the numerator.

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Step 3.2.3.4.6.1

Combine using the product rule for radicals.

$y = \pm \frac{7 \sqrt{6 \cdot 11}}{2 \cdot 6}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.6.2

Multiply $6$ by $11$ .

$y = \pm \frac{7 \sqrt{66}}{2 \cdot 6}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{66}}{2 \cdot 6}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.4.7

Multiply $2$ by $6$ .

$y = \pm \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \pm \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.5

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

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Step 3.2.3.5.1

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

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.5.2

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

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.2.3.5.3

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

$y = \frac{7 \sqrt{66}}{12}, - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \frac{7 \sqrt{66}}{12}, - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \frac{7 \sqrt{66}}{12}, - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + y) (6 - y)}$

$y = \frac{7 \sqrt{66}}{12}, - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + y) (6 - y)}$

Step 3.3

Replace all occurrences of $y$ with $\frac{7 \sqrt{66}}{12}$ in each equation.

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Step 3.3.1

Replace all occurrences of $y$ in $x = - \sqrt{(6 + y) (6 - y)}$ with $\frac{7 \sqrt{66}}{12}$ .

$x = - \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2

Simplify $x = - \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$ .

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Remove parentheses.

$x = - \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Simplify $- \sqrt{(6 + \frac{7 \sqrt{66}}{12}) (6 - (\frac{7 \sqrt{66}}{12}))}$ .

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Step 3.3.2.2.1.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = - \sqrt{(6 \cdot \frac{12}{12} + \frac{7 \sqrt{66}}{12}) (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.2

Combine $6$ and $\frac{12}{12}$ .

$x = - \sqrt{(\frac{6 \cdot 12}{12} + \frac{7 \sqrt{66}}{12}) (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.3

Simplify the expression.

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Step 3.3.2.2.1.3.1

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{6 \cdot 12 + 7 \sqrt{66}}{12} \cdot (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.3.2

Multiply $6$ by $12$ .

$x = - \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot (6 - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.4

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = - \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot (6 \cdot \frac{12}{12} - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.5

Combine fractions.

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Step 3.3.2.2.1.5.1

Combine $6$ and $\frac{12}{12}$ .

$x = - \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot (\frac{6 \cdot 12}{12} - \frac{7 \sqrt{66}}{12})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.5.2

Simplify the expression.

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Step 3.3.2.2.1.5.2.1

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot \frac{6 \cdot 12 - 7 \sqrt{66}}{12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.5.2.2

Multiply $6$ by $12$ .

$x = - \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot \frac{72 - 7 \sqrt{66}}{12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{72 + 7 \sqrt{66}}{12} \cdot \frac{72 - 7 \sqrt{66}}{12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.5.3

Multiply $\frac{72 + 7 \sqrt{66}}{12}$ by $\frac{72 - 7 \sqrt{66}}{12}$ .

$x = - \sqrt{\frac{(72 + 7 \sqrt{66}) (72 - 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{(72 + 7 \sqrt{66}) (72 - 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6

Simplify the numerator.

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Step 3.3.2.2.1.6.1

Expand $(72 + 7 \sqrt{66}) (72 - 7 \sqrt{66})$ using the FOIL Method.

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Step 3.3.2.2.1.6.1.1

Apply the distributive property.

$x = - \sqrt{\frac{72 (72 - 7 \sqrt{66}) + 7 \sqrt{66} (72 - 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.1.2

Apply the distributive property.

$x = - \sqrt{\frac{72 \cdot 72 + 72 (- 7 \sqrt{66}) + 7 \sqrt{66} (72 - 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.1.3

Apply the distributive property.

$x = - \sqrt{\frac{72 \cdot 72 + 72 (- 7 \sqrt{66}) + 7 \sqrt{66} \cdot 72 + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{72 \cdot 72 + 72 (- 7 \sqrt{66}) + 7 \sqrt{66} \cdot 72 + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2

Simplify and combine like terms.

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Step 3.3.2.2.1.6.2.1

Simplify each term.

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Step 3.3.2.2.1.6.2.1.1

Multiply $72$ by $72$ .

$x = - \sqrt{\frac{5184 + 72 (- 7 \sqrt{66}) + 7 \sqrt{66} \cdot 72 + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.2

Multiply $- 7$ by $72$ .

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 7 \sqrt{66} \cdot 72 + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.3

Multiply $72$ by $7$ .

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} + 7 \sqrt{66} (- 7 \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.4

Multiply $7 \sqrt{66} (- 7 \sqrt{66})$ .

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Step 3.3.2.2.1.6.2.1.4.1

Multiply $- 7$ by $7$ .

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \sqrt{66} \sqrt{66}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.4.2

Raise $\sqrt{66}$ to the power of $1$ .

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 (\sqrt{66} \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.4.3

Raise $\sqrt{66}$ to the power of $1$ .

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 (\sqrt{66} \sqrt{66})}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.4.4

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

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 {\sqrt{66}}^{1 + 1}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.4.5

Add $1$ and $1$ .

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 {\sqrt{66}}^{2}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 {\sqrt{66}}^{2}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.5

Rewrite ${\sqrt{66}}^{2}$ as $66$ .

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Step 3.3.2.2.1.6.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{66}$ as $66^{\frac{1}{2}}$ .

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 {(66^{\frac{1}{2}})}^{2}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.5.2

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

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66^{\frac{1}{2} \cdot 2}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.5.3

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

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66^{\frac{2}{2}}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.6.2.1.5.4.1

Cancel the common factor.

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66^{\frac{2}{2}}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.5.4.2

Rewrite the expression.

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.5.5

Evaluate the exponent.

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.1.6

Multiply $- 49$ by $66$ .

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 3234}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{5184 - 504 \sqrt{66} + 504 \sqrt{66} - 3234}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.2

Subtract $3234$ from $5184$ .

$x = - \sqrt{\frac{1950 - 504 \sqrt{66} + 504 \sqrt{66}}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.3

Add $- 504 \sqrt{66}$ and $504 \sqrt{66}$ .

$x = - \sqrt{\frac{1950 + 0}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.6.2.4

Add $1950$ and $0$ .

$x = - \sqrt{\frac{1950}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{1950}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{1950}{12 \cdot 12}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.7

Reduce the expression by cancelling the common factors.

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Step 3.3.2.2.1.7.1

Multiply $12$ by $12$ .

$x = - \sqrt{\frac{1950}{144}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.7.2

Cancel the common factor of $1950$ and $144$ .

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Step 3.3.2.2.1.7.2.1

Factor $6$ out of $1950$ .

$x = - \sqrt{\frac{6 (325)}{144}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.7.2.2

Cancel the common factors.

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Step 3.3.2.2.1.7.2.2.1

Factor $6$ out of $144$ .

$x = - \sqrt{\frac{6 \cdot 325}{6 \cdot 24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.7.2.2.2

Cancel the common factor.

$x = - \sqrt{\frac{6 \cdot 325}{6 \cdot 24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.7.2.2.3

Rewrite the expression.

$x = - \sqrt{\frac{325}{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{325}{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{325}{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{325}{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.8

Rewrite $\sqrt{\frac{325}{24}}$ as $\frac{\sqrt{325}}{\sqrt{24}}$ .

$x = - \frac{\sqrt{325}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.9

Simplify the numerator.

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Step 3.3.2.2.1.9.1

Rewrite $325$ as $5^{2} \cdot 13$ .

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Step 3.3.2.2.1.9.1.1

Factor $25$ out of $325$ .

$x = - \frac{\sqrt{25 (13)}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.9.1.2

Rewrite $25$ as $5^{2}$ .

$x = - \frac{\sqrt{5^{2} \cdot 13}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{\sqrt{5^{2} \cdot 13}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.9.2

Pull terms out from under the radical.

$x = - \frac{5 \sqrt{13}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13}}{\sqrt{24}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.10

Simplify the denominator.

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Step 3.3.2.2.1.10.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 3.3.2.2.1.10.1.1

Factor $4$ out of $24$ .

$x = - \frac{5 \sqrt{13}}{\sqrt{4 (6)}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.10.1.2

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

$x = - \frac{5 \sqrt{13}}{\sqrt{2^{2} \cdot 6}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13}}{\sqrt{2^{2} \cdot 6}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.10.2

Pull terms out from under the radical.

$x = - \frac{5 \sqrt{13}}{2 \sqrt{6}}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13}}{2 \sqrt{6}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.11

Multiply $\frac{5 \sqrt{13}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = - (\frac{5 \sqrt{13}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12

Combine and simplify the denominator.

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Step 3.3.2.2.1.12.1

Multiply $\frac{5 \sqrt{13}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.2

Move $\sqrt{6}$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.3

Raise $\sqrt{6}$ to the power of $1$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.4

Raise $\sqrt{6}$ to the power of $1$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.5

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

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.6

Add $1$ and $1$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 {\sqrt{6}}^{2}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.3.2.2.1.12.7.1

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

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.7.2

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

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.7.3

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

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.7.4

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.12.7.4.1

Cancel the common factor.

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.7.4.2

Rewrite the expression.

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.12.7.5

Evaluate the exponent.

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.13

Simplify the numerator.

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Step 3.3.2.2.1.13.1

Combine using the product rule for radicals.

$x = - \frac{5 \sqrt{6 \cdot 13}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.13.2

Multiply $6$ by $13$ .

$x = - \frac{5 \sqrt{78}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{2 \cdot 6}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.3.2.2.1.14

Multiply $2$ by $6$ .

$x = - \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = \frac{7 \sqrt{66}}{12}$

Step 3.4

Replace all occurrences of $y$ with $- \frac{7 \sqrt{66}}{12}$ in each equation.

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Step 3.4.1

Replace all occurrences of $y$ in $x = - \sqrt{(6 + y) (6 - y)}$ with $- \frac{7 \sqrt{66}}{12}$ .

$x = - \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2

Simplify $x = - \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$ .

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Step 3.4.2.1

Simplify the left side.

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Step 3.4.2.1.1

Remove parentheses.

$x = - \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2

Simplify the right side.

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Step 3.4.2.2.1

Simplify $- \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 - (- \frac{7 \sqrt{66}}{12}))}$ .

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Step 3.4.2.2.1.1

Multiply $- - \frac{7 \sqrt{66}}{12}$ .

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Step 3.4.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$x = - \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 + 1 (\frac{7 \sqrt{66}}{12}))}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.1.2

Multiply $\frac{7 \sqrt{66}}{12}$ by $1$ .

$x = - \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{(6 - \frac{7 \sqrt{66}}{12}) (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = - \sqrt{(6 \cdot \frac{12}{12} - \frac{7 \sqrt{66}}{12}) (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.3

Combine $6$ and $\frac{12}{12}$ .

$x = - \sqrt{(\frac{6 \cdot 12}{12} - \frac{7 \sqrt{66}}{12}) (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.4

Simplify the expression.

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Step 3.4.2.2.1.4.1

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{6 \cdot 12 - 7 \sqrt{66}}{12} \cdot (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.4.2

Multiply $6$ by $12$ .

$x = - \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot (6 + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.5

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = - \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot (6 \cdot \frac{12}{12} + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.6

Combine fractions.

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Step 3.4.2.2.1.6.1

Combine $6$ and $\frac{12}{12}$ .

$x = - \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot (\frac{6 \cdot 12}{12} + \frac{7 \sqrt{66}}{12})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.6.2

Simplify the expression.

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Step 3.4.2.2.1.6.2.1

Combine the numerators over the common denominator.

$x = - \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot \frac{6 \cdot 12 + 7 \sqrt{66}}{12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.6.2.2

Multiply $6$ by $12$ .

$x = - \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot \frac{72 + 7 \sqrt{66}}{12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{72 - 7 \sqrt{66}}{12} \cdot \frac{72 + 7 \sqrt{66}}{12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.6.3

Multiply $\frac{72 - 7 \sqrt{66}}{12}$ by $\frac{72 + 7 \sqrt{66}}{12}$ .

$x = - \sqrt{\frac{(72 - 7 \sqrt{66}) (72 + 7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{(72 - 7 \sqrt{66}) (72 + 7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7

Simplify the numerator.

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Step 3.4.2.2.1.7.1

Expand $(72 - 7 \sqrt{66}) (72 + 7 \sqrt{66})$ using the FOIL Method.

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Step 3.4.2.2.1.7.1.1

Apply the distributive property.

$x = - \sqrt{\frac{72 (72 + 7 \sqrt{66}) - 7 \sqrt{66} (72 + 7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.1.2

Apply the distributive property.

$x = - \sqrt{\frac{72 \cdot 72 + 72 (7 \sqrt{66}) - 7 \sqrt{66} (72 + 7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.1.3

Apply the distributive property.

$x = - \sqrt{\frac{72 \cdot 72 + 72 (7 \sqrt{66}) - 7 \sqrt{66} \cdot 72 - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{72 \cdot 72 + 72 (7 \sqrt{66}) - 7 \sqrt{66} \cdot 72 - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2

Simplify and combine like terms.

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Step 3.4.2.2.1.7.2.1

Simplify each term.

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Step 3.4.2.2.1.7.2.1.1

Multiply $72$ by $72$ .

$x = - \sqrt{\frac{5184 + 72 (7 \sqrt{66}) - 7 \sqrt{66} \cdot 72 - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.2

Multiply $7$ by $72$ .

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 7 \sqrt{66} \cdot 72 - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.3

Multiply $72$ by $- 7$ .

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 7 \sqrt{66} (7 \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.4

Multiply $- 7 \sqrt{66} (7 \sqrt{66})$ .

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Step 3.4.2.2.1.7.2.1.4.1

Multiply $7$ by $- 7$ .

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \sqrt{66} \sqrt{66}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.4.2

Raise $\sqrt{66}$ to the power of $1$ .

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 (\sqrt{66} \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.4.3

Raise $\sqrt{66}$ to the power of $1$ .

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 (\sqrt{66} \sqrt{66})}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.4.4

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

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 {\sqrt{66}}^{1 + 1}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.4.5

Add $1$ and $1$ .

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 {\sqrt{66}}^{2}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 {\sqrt{66}}^{2}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.5

Rewrite ${\sqrt{66}}^{2}$ as $66$ .

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Step 3.4.2.2.1.7.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{66}$ as $66^{\frac{1}{2}}$ .

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 {(66^{\frac{1}{2}})}^{2}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.5.2

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

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66^{\frac{1}{2} \cdot 2}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.5.3

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

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66^{\frac{2}{2}}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.5.4

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.7.2.1.5.4.1

Cancel the common factor.

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66^{\frac{2}{2}}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.5.4.2

Rewrite the expression.

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.5.5

Evaluate the exponent.

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 49 \cdot 66}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.1.6

Multiply $- 49$ by $66$ .

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 3234}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{5184 + 504 \sqrt{66} - 504 \sqrt{66} - 3234}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.2

Subtract $3234$ from $5184$ .

$x = - \sqrt{\frac{1950 + 504 \sqrt{66} - 504 \sqrt{66}}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.3

Subtract $504 \sqrt{66}$ from $504 \sqrt{66}$ .

$x = - \sqrt{\frac{1950 + 0}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.7.2.4

Add $1950$ and $0$ .

$x = - \sqrt{\frac{1950}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{1950}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{1950}{12 \cdot 12}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.8

Reduce the expression by cancelling the common factors.

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Step 3.4.2.2.1.8.1

Multiply $12$ by $12$ .

$x = - \sqrt{\frac{1950}{144}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.8.2

Cancel the common factor of $1950$ and $144$ .

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Step 3.4.2.2.1.8.2.1

Factor $6$ out of $1950$ .

$x = - \sqrt{\frac{6 (325)}{144}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.8.2.2

Cancel the common factors.

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Step 3.4.2.2.1.8.2.2.1

Factor $6$ out of $144$ .

$x = - \sqrt{\frac{6 \cdot 325}{6 \cdot 24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.8.2.2.2

Cancel the common factor.

$x = - \sqrt{\frac{6 \cdot 325}{6 \cdot 24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.8.2.2.3

Rewrite the expression.

$x = - \sqrt{\frac{325}{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{325}{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{325}{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \sqrt{\frac{325}{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.9

Rewrite $\sqrt{\frac{325}{24}}$ as $\frac{\sqrt{325}}{\sqrt{24}}$ .

$x = - \frac{\sqrt{325}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.10

Simplify the numerator.

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Step 3.4.2.2.1.10.1

Rewrite $325$ as $5^{2} \cdot 13$ .

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Step 3.4.2.2.1.10.1.1

Factor $25$ out of $325$ .

$x = - \frac{\sqrt{25 (13)}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.10.1.2

Rewrite $25$ as $5^{2}$ .

$x = - \frac{\sqrt{5^{2} \cdot 13}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{\sqrt{5^{2} \cdot 13}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.10.2

Pull terms out from under the radical.

$x = - \frac{5 \sqrt{13}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13}}{\sqrt{24}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.11

Simplify the denominator.

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Step 3.4.2.2.1.11.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 3.4.2.2.1.11.1.1

Factor $4$ out of $24$ .

$x = - \frac{5 \sqrt{13}}{\sqrt{4 (6)}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.11.1.2

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

$x = - \frac{5 \sqrt{13}}{\sqrt{2^{2} \cdot 6}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13}}{\sqrt{2^{2} \cdot 6}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.11.2

Pull terms out from under the radical.

$x = - \frac{5 \sqrt{13}}{2 \sqrt{6}}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13}}{2 \sqrt{6}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.12

Multiply $\frac{5 \sqrt{13}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = - (\frac{5 \sqrt{13}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13

Combine and simplify the denominator.

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Step 3.4.2.2.1.13.1

Multiply $\frac{5 \sqrt{13}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.2

Move $\sqrt{6}$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.3

Raise $\sqrt{6}$ to the power of $1$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.4

Raise $\sqrt{6}$ to the power of $1$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.5

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

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.6

Add $1$ and $1$ .

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 {\sqrt{6}}^{2}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.4.2.2.1.13.7.1

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

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.7.2

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

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.7.3

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

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.7.4

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.13.7.4.1

Cancel the common factor.

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.7.4.2

Rewrite the expression.

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.13.7.5

Evaluate the exponent.

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{13} \sqrt{6}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.14

Simplify the numerator.

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Step 3.4.2.2.1.14.1

Combine using the product rule for radicals.

$x = - \frac{5 \sqrt{6 \cdot 13}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.14.2

Multiply $6$ by $13$ .

$x = - \frac{5 \sqrt{78}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{2 \cdot 6}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 3.4.2.2.1.15

Multiply $2$ by $6$ .

$x = - \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}$

$y = - \frac{7 \sqrt{66}}{12}$

Step 4

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

$(\frac{5 \sqrt{78}}{12}, \frac{7 \sqrt{66}}{12})$

$(\frac{5 \sqrt{78}}{12}, - \frac{7 \sqrt{66}}{12})$

$(- \frac{5 \sqrt{78}}{12}, \frac{7 \sqrt{66}}{12})$

$(- \frac{5 \sqrt{78}}{12}, - \frac{7 \sqrt{66}}{12})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(\frac{5 \sqrt{78}}{12}, \frac{7 \sqrt{66}}{12}), (\frac{5 \sqrt{78}}{12}, - \frac{7 \sqrt{66}}{12}), (- \frac{5 \sqrt{78}}{12}, \frac{7 \sqrt{66}}{12}), (- \frac{5 \sqrt{78}}{12}, - \frac{7 \sqrt{66}}{12})$

Equation Form:

$x = \frac{5 \sqrt{78}}{12}, y = \frac{7 \sqrt{66}}{12}$

$x = \frac{5 \sqrt{78}}{12}, y = - \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}, y = \frac{7 \sqrt{66}}{12}$

$x = - \frac{5 \sqrt{78}}{12}, y = - \frac{7 \sqrt{66}}{12}$

Step 6