Algebra Examples

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$ ${(y + 2)}^{2} = 8 (x - 9)$

Step 1

Solve for $y$ in ${(y + 2)}^{2} = 8 (x - 9)$ .

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Step 1.1

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

$y + 2 = \pm \sqrt{8 (x - 9)}$

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.2

Simplify $\pm \sqrt{8 (x - 9)}$ .

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Step 1.2.1

Rewrite $8 (x - 9)$ as $2^{2} \cdot (2 (x - 9))$ .

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Step 1.2.1.1

Factor $4$ out of $8$ .

$y + 2 = \pm \sqrt{4 (2) (x - 9)}$

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.2.1.2

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

$y + 2 = \pm \sqrt{2^{2} \cdot (2 (x - 9))}$

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.2.1.3

Add parentheses.

$y + 2 = \pm \sqrt{2^{2} \cdot (2 (x - 9))}$

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

$y + 2 = \pm \sqrt{2^{2} \cdot (2 (x - 9))}$

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.2.2

Pull terms out from under the radical.

$y + 2 = \pm 2 \sqrt{2 (x - 9)}$

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

$y + 2 = \pm 2 \sqrt{2 (x - 9)}$

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.3

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

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Step 1.3.1

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

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

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.3.2

Subtract $2$ from both sides of the equation.

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

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.3.3

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

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

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.3.4

Subtract $2$ from both sides of the equation.

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

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 1.3.5

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

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

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

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

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

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

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

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

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

${(x - 3)}^{2} + {(y + 2)}^{2} = 36$

Step 2

Solve the system $y = 2 \sqrt{2 (x - 9)} - 2, {(x - 3)}^{2} + {(y + 2)}^{2} = 36$ .

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Step 2.1

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

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Step 2.1.1

Replace all occurrences of $y$ in ${(x - 3)}^{2} + {(y + 2)}^{2} = 36$ with $2 \sqrt{2 (x - 9)} - 2$ .

${(x - 3)}^{2} + {((2 \sqrt{2 (x - 9)} - 2) + 2)}^{2} = 36$

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

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

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Step 2.1.2.1.1

Combine the opposite terms in ${(x - 3)}^{2} + {((2 \sqrt{2 (x - 9)} - 2) + 2)}^{2}$ .

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Step 2.1.2.1.1.1

Add $- 2$ and $2$ .

${(x - 3)}^{2} + {(2 \sqrt{2 (x - 9)} + 0)}^{2} = 36$

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

Step 2.1.2.1.1.2

Add $2 \sqrt{2 (x - 9)}$ and $0$ .

${(x - 3)}^{2} + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

${(x - 3)}^{2} + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2

Simplify each term.

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Step 2.1.2.1.2.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$(x - 3) (x - 3) + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 2.1.2.1.2.2.1

Apply the distributive property.

$x (x - 3) - 3 (x - 3) + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 (x - 3) + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.1.2.3.1

Simplify each term.

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Step 2.1.2.1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2.3.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2.3.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

$x^{2} - 3 x - 3 x + 9 + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 6 x + 9 + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

$x^{2} - 6 x + 9 + {(2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 2.1.2.1.2.4

Apply the product rule to $2 \sqrt{2 (x - 9)}$ .

$x^{2} - 6 x + 9 + 2^{2} {\sqrt{2 (x - 9)}}^{2} = 36$

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

Step 2.1.2.1.2.5

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

$x^{2} - 6 x + 9 + 4 {\sqrt{2 (x - 9)}}^{2} = 36$

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

Step 2.1.2.1.2.6

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

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Step 2.1.2.1.2.6.1

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

$x^{2} - 6 x + 9 + 4 {({(2 (x - 9))}^{\frac{1}{2}})}^{2} = 36$

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

Step 2.1.2.1.2.6.2

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

$x^{2} - 6 x + 9 + 4 {(2 (x - 9))}^{\frac{1}{2} \cdot 2} = 36$

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

Step 2.1.2.1.2.6.3

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

$x^{2} - 6 x + 9 + 4 {(2 (x - 9))}^{\frac{2}{2}} = 36$

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

Step 2.1.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.2.6.4.1

Cancel the common factor.

$x^{2} - 6 x + 9 + 4 {(2 (x - 9))}^{\frac{2}{2}} = 36$

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

Step 2.1.2.1.2.6.4.2

Rewrite the expression.

$x^{2} - 6 x + 9 + 4 (2 (x - 9)) = 36$

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

$x^{2} - 6 x + 9 + 4 (2 (x - 9)) = 36$

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

Step 2.1.2.1.2.6.5

Simplify.

$x^{2} - 6 x + 9 + 4 (2 (x - 9)) = 36$

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

$x^{2} - 6 x + 9 + 4 (2 (x - 9)) = 36$

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

Step 2.1.2.1.2.7

Apply the distributive property.

$x^{2} - 6 x + 9 + 4 (2 x + 2 \cdot - 9) = 36$

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

Step 2.1.2.1.2.8

Multiply $2$ by $- 9$ .

$x^{2} - 6 x + 9 + 4 (2 x - 18) = 36$

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

Step 2.1.2.1.2.9

Apply the distributive property.

$x^{2} - 6 x + 9 + 4 (2 x) + 4 \cdot - 18 = 36$

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

Step 2.1.2.1.2.10

Multiply $2$ by $4$ .

$x^{2} - 6 x + 9 + 8 x + 4 \cdot - 18 = 36$

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

Step 2.1.2.1.2.11

Multiply $4$ by $- 18$ .

$x^{2} - 6 x + 9 + 8 x - 72 = 36$

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

$x^{2} - 6 x + 9 + 8 x - 72 = 36$

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

Step 2.1.2.1.3

Simplify by adding terms.

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Step 2.1.2.1.3.1

Add $- 6 x$ and $8 x$ .

$x^{2} + 2 x + 9 - 72 = 36$

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

Step 2.1.2.1.3.2

Subtract $72$ from $9$ .

$x^{2} + 2 x - 63 = 36$

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

$x^{2} + 2 x - 63 = 36$

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

$x^{2} + 2 x - 63 = 36$

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

$x^{2} + 2 x - 63 = 36$

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

$x^{2} + 2 x - 63 = 36$

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

Step 2.2

Solve for $x$ in $x^{2} + 2 x - 63 = 36$ .

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Step 2.2.1

Subtract $36$ from both sides of the equation.

$x^{2} + 2 x - 63 - 36 = 0$

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

Step 2.2.2

Subtract $36$ from $- 63$ .

$x^{2} + 2 x - 99 = 0$

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

Step 2.2.3

Factor $x^{2} + 2 x - 99$ using the AC method.

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Step 2.2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 99$ and whose sum is $2$ .

$- 9, 11$

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

Step 2.2.3.2

Write the factored form using these integers.

$(x - 9) (x + 11) = 0$

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

$(x - 9) (x + 11) = 0$

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

Step 2.2.4

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

$x - 9 = 0$

$x + 11 = 0$

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

Step 2.2.5

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

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

Step 2.2.5.2

Add $9$ to both sides of the equation.

$x = 9$

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

$x = 9$

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

Step 2.2.6

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 2.2.6.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

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

Step 2.2.6.2

Subtract $11$ from both sides of the equation.

$x = - 11$

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

$x = - 11$

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

Step 2.2.7

The final solution is all the values that make $(x - 9) (x + 11) = 0$ true.

$x = 9, - 11$

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

$x = 9, - 11$

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

Step 2.3

Replace all occurrences of $x$ with $9$ in each equation.

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Step 2.3.1

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

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

$x = 9$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Simplify each term.

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Step 2.3.2.1.1.1

Subtract $9$ from $9$ .

$y = 2 \sqrt{2 \cdot 0} - 2$

$x = 9$

Step 2.3.2.1.1.2

Multiply $2$ by $0$ .

$y = 2 \sqrt{0} - 2$

$x = 9$

Step 2.3.2.1.1.3

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

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

$x = 9$

Step 2.3.2.1.1.4

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

$y = 2 \cdot 0 - 2$

$x = 9$

Step 2.3.2.1.1.5

Multiply $2$ by $0$ .

$y = 0 - 2$

$x = 9$

$y = 0 - 2$

$x = 9$

Step 2.3.2.1.2

Subtract $2$ from $0$ .

$y = - 2$

$x = 9$

$y = - 2$

$x = 9$

$y = - 2$

$x = 9$

$y = - 2$

$x = 9$

Step 2.4

Replace all occurrences of $x$ with $- 11$ in each equation.

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Step 2.4.1

Replace all occurrences of $x$ in $y = 2 \sqrt{2 (x - 9)} - 2$ with $- 11$ .

$y = 2 \sqrt{2 ((- 11) - 9)} - 2$

$x = - 11$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $2 \sqrt{2 ((- 11) - 9)} - 2$ .

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Step 2.4.2.1.1

Simplify each term.

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Step 2.4.2.1.1.1

Subtract $9$ from $- 11$ .

$y = 2 \sqrt{2 \cdot - 20} - 2$

$x = - 11$

Step 2.4.2.1.1.2

Multiply $2$ by $- 20$ .

$y = 2 \sqrt{- 40} - 2$

$x = - 11$

Step 2.4.2.1.1.3

Rewrite $- 40$ as $- 1 (40)$ .

$y = 2 \sqrt{- 1 \cdot 40} - 2$

$x = - 11$

Step 2.4.2.1.1.4

Rewrite $\sqrt{- 1 (40)}$ as $\sqrt{- 1} \cdot \sqrt{40}$ .

$y = 2 (\sqrt{- 1} \cdot \sqrt{40}) - 2$

$x = - 11$

Step 2.4.2.1.1.5

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

$y = 2 (i \cdot \sqrt{40}) - 2$

$x = - 11$

Step 2.4.2.1.1.6

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 2.4.2.1.1.6.1

Factor $4$ out of $40$ .

$y = 2 (i \cdot \sqrt{4 (10)}) - 2$

$x = - 11$

Step 2.4.2.1.1.6.2

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

$y = 2 (i \cdot \sqrt{2^{2} \cdot 10}) - 2$

$x = - 11$

$y = 2 (i \cdot \sqrt{2^{2} \cdot 10}) - 2$

$x = - 11$

Step 2.4.2.1.1.7

Pull terms out from under the radical.

$y = 2 (i \cdot (2 \sqrt{10})) - 2$

$x = - 11$

Step 2.4.2.1.1.8

Move $2$ to the left of $i$ .

$y = 2 (2 \cdot (i \sqrt{10})) - 2$

$x = - 11$

Step 2.4.2.1.1.9

Multiply $2$ by $2$ .

$y = 4 i \sqrt{10} - 2$

$x = - 11$

$y = 4 i \sqrt{10} - 2$

$x = - 11$

Step 2.4.2.1.2

Reorder $4 i \sqrt{10}$ and $- 2$ .

$y = - 2 + 4 i \sqrt{10}$

$x = - 11$

$y = - 2 + 4 i \sqrt{10}$

$x = - 11$

$y = - 2 + 4 i \sqrt{10}$

$x = - 11$

$y = - 2 + 4 i \sqrt{10}$

$x = - 11$

$y = - 2 + 4 i \sqrt{10}$

$x = - 11$

Step 3

Solve the system $y = - 2 \sqrt{2 (x - 9)} - 2, {(x - 3)}^{2} + {(y + 2)}^{2} = 36$ .

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Step 3.1

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

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Step 3.1.1

Replace all occurrences of $y$ in ${(x - 3)}^{2} + {(y + 2)}^{2} = 36$ with $- 2 \sqrt{2 (x - 9)} - 2$ .

${(x - 3)}^{2} + {((- 2 \sqrt{2 (x - 9)} - 2) + 2)}^{2} = 36$

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

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

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Step 3.1.2.1.1

Combine the opposite terms in ${(x - 3)}^{2} + {((- 2 \sqrt{2 (x - 9)} - 2) + 2)}^{2}$ .

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Step 3.1.2.1.1.1

Add $- 2$ and $2$ .

${(x - 3)}^{2} + {(- 2 \sqrt{2 (x - 9)} + 0)}^{2} = 36$

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

Step 3.1.2.1.1.2

Add $- 2 \sqrt{2 (x - 9)}$ and $0$ .

${(x - 3)}^{2} + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

${(x - 3)}^{2} + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2

Simplify each term.

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Step 3.1.2.1.2.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$(x - 3) (x - 3) + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 3.1.2.1.2.2.1

Apply the distributive property.

$x (x - 3) - 3 (x - 3) + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 (x - 3) + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2.3

Simplify and combine like terms.

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Step 3.1.2.1.2.3.1

Simplify each term.

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Step 3.1.2.1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2.3.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2.3.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

$x^{2} - 3 x - 3 x + 9 + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 6 x + 9 + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

$x^{2} - 6 x + 9 + {(- 2 \sqrt{2 (x - 9)})}^{2} = 36$

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

Step 3.1.2.1.2.4

Apply the product rule to $- 2 \sqrt{2 (x - 9)}$ .

$x^{2} - 6 x + 9 + {(- 2)}^{2} {\sqrt{2 (x - 9)}}^{2} = 36$

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

Step 3.1.2.1.2.5

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

$x^{2} - 6 x + 9 + 4 {\sqrt{2 (x - 9)}}^{2} = 36$

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

Step 3.1.2.1.2.6

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

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Step 3.1.2.1.2.6.1

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

$x^{2} - 6 x + 9 + 4 {({(2 (x - 9))}^{\frac{1}{2}})}^{2} = 36$

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

Step 3.1.2.1.2.6.2

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

$x^{2} - 6 x + 9 + 4 {(2 (x - 9))}^{\frac{1}{2} \cdot 2} = 36$

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

Step 3.1.2.1.2.6.3

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

$x^{2} - 6 x + 9 + 4 {(2 (x - 9))}^{\frac{2}{2}} = 36$

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

Step 3.1.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.2.6.4.1

Cancel the common factor.

$x^{2} - 6 x + 9 + 4 {(2 (x - 9))}^{\frac{2}{2}} = 36$

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

Step 3.1.2.1.2.6.4.2

Rewrite the expression.

$x^{2} - 6 x + 9 + 4 (2 (x - 9)) = 36$

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

$x^{2} - 6 x + 9 + 4 (2 (x - 9)) = 36$

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

Step 3.1.2.1.2.6.5

Simplify.

$x^{2} - 6 x + 9 + 4 (2 (x - 9)) = 36$

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

$x^{2} - 6 x + 9 + 4 (2 (x - 9)) = 36$

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

Step 3.1.2.1.2.7

Apply the distributive property.

$x^{2} - 6 x + 9 + 4 (2 x + 2 \cdot - 9) = 36$

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

Step 3.1.2.1.2.8

Multiply $2$ by $- 9$ .

$x^{2} - 6 x + 9 + 4 (2 x - 18) = 36$

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

Step 3.1.2.1.2.9

Apply the distributive property.

$x^{2} - 6 x + 9 + 4 (2 x) + 4 \cdot - 18 = 36$

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

Step 3.1.2.1.2.10

Multiply $2$ by $4$ .

$x^{2} - 6 x + 9 + 8 x + 4 \cdot - 18 = 36$

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

Step 3.1.2.1.2.11

Multiply $4$ by $- 18$ .

$x^{2} - 6 x + 9 + 8 x - 72 = 36$

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

$x^{2} - 6 x + 9 + 8 x - 72 = 36$

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

Step 3.1.2.1.3

Simplify by adding terms.

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Step 3.1.2.1.3.1

Add $- 6 x$ and $8 x$ .

$x^{2} + 2 x + 9 - 72 = 36$

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

Step 3.1.2.1.3.2

Subtract $72$ from $9$ .

$x^{2} + 2 x - 63 = 36$

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

$x^{2} + 2 x - 63 = 36$

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

$x^{2} + 2 x - 63 = 36$

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

$x^{2} + 2 x - 63 = 36$

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

$x^{2} + 2 x - 63 = 36$

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

Step 3.2

Solve for $x$ in $x^{2} + 2 x - 63 = 36$ .

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Step 3.2.1

Subtract $36$ from both sides of the equation.

$x^{2} + 2 x - 63 - 36 = 0$

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

Step 3.2.2

Subtract $36$ from $- 63$ .

$x^{2} + 2 x - 99 = 0$

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

Step 3.2.3

Factor $x^{2} + 2 x - 99$ using the AC method.

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Step 3.2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 99$ and whose sum is $2$ .

$- 9, 11$

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

Step 3.2.3.2

Write the factored form using these integers.

$(x - 9) (x + 11) = 0$

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

$(x - 9) (x + 11) = 0$

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

Step 3.2.4

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

$x - 9 = 0$

$x + 11 = 0$

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

Step 3.2.5

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 3.2.5.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

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

Step 3.2.5.2

Add $9$ to both sides of the equation.

$x = 9$

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

$x = 9$

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

Step 3.2.6

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 3.2.6.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

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

Step 3.2.6.2

Subtract $11$ from both sides of the equation.

$x = - 11$

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

$x = - 11$

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

Step 3.2.7

The final solution is all the values that make $(x - 9) (x + 11) = 0$ true.

$x = 9, - 11$

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

$x = 9, - 11$

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

Step 3.3

Replace all occurrences of $x$ with $9$ in each equation.

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Step 3.3.1

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

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

$x = 9$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Simplify each term.

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Step 3.3.2.1.1.1

Subtract $9$ from $9$ .

$y = - 2 \sqrt{2 \cdot 0} - 2$

$x = 9$

Step 3.3.2.1.1.2

Multiply $2$ by $0$ .

$y = - 2 \sqrt{0} - 2$

$x = 9$

Step 3.3.2.1.1.3

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

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

$x = 9$

Step 3.3.2.1.1.4

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

$y = - 2 \cdot 0 - 2$

$x = 9$

Step 3.3.2.1.1.5

Multiply $- 2$ by $0$ .

$y = 0 - 2$

$x = 9$

$y = 0 - 2$

$x = 9$

Step 3.3.2.1.2

Subtract $2$ from $0$ .

$y = - 2$

$x = 9$

$y = - 2$

$x = 9$

$y = - 2$

$x = 9$

$y = - 2$

$x = 9$

Step 3.4

Replace all occurrences of $x$ with $- 11$ in each equation.

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Step 3.4.1

Replace all occurrences of $x$ in $y = - 2 \sqrt{2 (x - 9)} - 2$ with $- 11$ .

$y = - 2 \sqrt{2 ((- 11) - 9)} - 2$

$x = - 11$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 2 \sqrt{2 ((- 11) - 9)} - 2$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Subtract $9$ from $- 11$ .

$y = - 2 \sqrt{2 \cdot - 20} - 2$

$x = - 11$

Step 3.4.2.1.1.2

Multiply $2$ by $- 20$ .

$y = - 2 \sqrt{- 40} - 2$

$x = - 11$

Step 3.4.2.1.1.3

Rewrite $- 40$ as $- 1 (40)$ .

$y = - 2 \sqrt{- 1 \cdot 40} - 2$

$x = - 11$

Step 3.4.2.1.1.4

Rewrite $\sqrt{- 1 (40)}$ as $\sqrt{- 1} \cdot \sqrt{40}$ .

$y = - 2 (\sqrt{- 1} \cdot \sqrt{40}) - 2$

$x = - 11$

Step 3.4.2.1.1.5

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

$y = - 2 (i \cdot \sqrt{40}) - 2$

$x = - 11$

Step 3.4.2.1.1.6

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 3.4.2.1.1.6.1

Factor $4$ out of $40$ .

$y = - 2 (i \cdot \sqrt{4 (10)}) - 2$

$x = - 11$

Step 3.4.2.1.1.6.2

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

$y = - 2 (i \cdot \sqrt{2^{2} \cdot 10}) - 2$

$x = - 11$

$y = - 2 (i \cdot \sqrt{2^{2} \cdot 10}) - 2$

$x = - 11$

Step 3.4.2.1.1.7

Pull terms out from under the radical.

$y = - 2 (i \cdot (2 \sqrt{10})) - 2$

$x = - 11$

Step 3.4.2.1.1.8

Move $2$ to the left of $i$ .

$y = - 2 (2 \cdot (i \sqrt{10})) - 2$

$x = - 11$

Step 3.4.2.1.1.9

Multiply $2$ by $- 2$ .

$y = - 4 i \sqrt{10} - 2$

$x = - 11$

$y = - 4 i \sqrt{10} - 2$

$x = - 11$

Step 3.4.2.1.2

Reorder $- 4 i \sqrt{10}$ and $- 2$ .

$y = - 2 - 4 i \sqrt{10}$

$x = - 11$

$y = - 2 - 4 i \sqrt{10}$

$x = - 11$

$y = - 2 - 4 i \sqrt{10}$

$x = - 11$

$y = - 2 - 4 i \sqrt{10}$

$x = - 11$

$y = - 2 - 4 i \sqrt{10}$

$x = - 11$

Step 4

List all of the solutions.

$y = - 2, x = 9$

$y = - 2 + 4 i \sqrt{10}, x = - 11$

$y = - 2 - 4 i \sqrt{10}, x = - 11$

Step 5