Finite Math Examples

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

Step 1

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

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Step 1.1

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

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

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

Step 1.2

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

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

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

Step 1.3

Simplify $\pm \sqrt{9 - y^{2}}$ .

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Step 1.3.1

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

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

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

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 $a = 3$ and $b = y$ .

$x = \pm \sqrt{(3 + y) (3 - y)}$

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

$x = \pm \sqrt{(3 + y) (3 - y)}$

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

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.

$x = \sqrt{(3 + y) (3 - y)}$

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

Step 1.4.2

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

$x = - \sqrt{(3 + y) (3 - y)}$

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

Step 1.4.3

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

$x = \sqrt{(3 + y) (3 - y)}$

$x = - \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

$x = - \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

$x = - \sqrt{(3 + y) (3 - y)}$

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

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

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

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

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.2

Expand $(\sqrt{(3 + y) (3 - y)} - 3) (\sqrt{(3 + y) (3 - y)} - 3)$ using the FOIL Method.

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Step 2.1.2.1.1.2.1

Apply the distributive property.

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.2.2

Apply the distributive property.

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.2.3

Apply the distributive property.

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

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 $\sqrt{(3 + y) (3 - y)} \sqrt{(3 + y) (3 - y)}$ .

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Step 2.1.2.1.1.3.1.1.1

Raise $\sqrt{(3 + y) (3 - y)}$ to the power of $1$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.1.2

Raise $\sqrt{(3 + y) (3 - y)}$ to the power of $1$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.1.3

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

${\sqrt{(3 + y) (3 - y)}}^{1 + 1} + \sqrt{(3 + y) (3 - y)} \cdot - 3 - 3 \sqrt{(3 + y) (3 - y)} - 3 \cdot - 3 + {(y + 3)}^{2} = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.1.4

Add $1$ and $1$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.2

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

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Step 2.1.2.1.1.3.1.2.1

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

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.2.2

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

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.2.3

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

${((3 + y) (3 - y))}^{\frac{2}{2}} + \sqrt{(3 + y) (3 - y)} \cdot - 3 - 3 \sqrt{(3 + y) (3 - y)} - 3 \cdot - 3 + {(y + 3)}^{2} = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.3.1.2.4.1

Cancel the common factor.

${((3 + y) (3 - y))}^{\frac{2}{2}} + \sqrt{(3 + y) (3 - y)} \cdot - 3 - 3 \sqrt{(3 + y) (3 - y)} - 3 \cdot - 3 + {(y + 3)}^{2} = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.2.4.2

Rewrite the expression.

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.2.5

Simplify.

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.3

Expand $(3 + y) (3 - y)$ using the FOIL Method.

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Step 2.1.2.1.1.3.1.3.1

Apply the distributive property.

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.3.2

Apply the distributive property.

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.3.3

Apply the distributive property.

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.4

Simplify and combine like terms.

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Step 2.1.2.1.1.3.1.4.1

Simplify each term.

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Step 2.1.2.1.1.3.1.4.1.1

Multiply $3$ by $3$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.4.1.2

Multiply $- 1$ by $3$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.4.1.3

Move $3$ to the left of $y$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.4.1.4

Rewrite using the commutative property of multiplication.

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.4.1.5

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

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Step 2.1.2.1.1.3.1.4.1.5.1

Move $y$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.4.1.5.2

Multiply $y$ by $y$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.4.2

Add $- 3 y$ and $3 y$ .

$9 + 0 - y^{2} + \sqrt{(3 + y) (3 - y)} \cdot - 3 - 3 \sqrt{(3 + y) (3 - y)} - 3 \cdot - 3 + {(y + 3)}^{2} = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.4.3

Add $9$ and $0$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.5

Move $- 3$ to the left of $\sqrt{(3 + y) (3 - y)}$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.1.6

Multiply $- 3$ by $- 3$ .

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

$x = \sqrt{(3 + y) (3 - y)}$

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.2

Add $9$ and $9$ .

$18 - y^{2} - 3 \sqrt{(3 + y) (3 - y)} - 3 \sqrt{(3 + y) (3 - y)} + {(y + 3)}^{2} = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.3.3

Subtract $3 \sqrt{(3 + y) (3 - y)}$ from $- 3 \sqrt{(3 + y) (3 - y)}$ .

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + {(y + 3)}^{2} = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + {(y + 3)}^{2} = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.4

Rewrite ${(y + 3)}^{2}$ as $(y + 3) (y + 3)$ .

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + (y + 3) (y + 3) = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.5

Expand $(y + 3) (y + 3)$ using the FOIL Method.

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Step 2.1.2.1.1.5.1

Apply the distributive property.

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y (y + 3) + 3 (y + 3) = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.5.2

Apply the distributive property.

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y \cdot y + y \cdot 3 + 3 (y + 3) = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.5.3

Apply the distributive property.

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y \cdot y + y \cdot 3 + 3 y + 3 \cdot 3 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y \cdot y + y \cdot 3 + 3 y + 3 \cdot 3 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.6

Simplify and combine like terms.

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Step 2.1.2.1.1.6.1

Simplify each term.

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Step 2.1.2.1.1.6.1.1

Multiply $y$ by $y$ .

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y^{2} + y \cdot 3 + 3 y + 3 \cdot 3 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.6.1.2

Move $3$ to the left of $y$ .

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y^{2} + 3 \cdot y + 3 y + 3 \cdot 3 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.6.1.3

Multiply $3$ by $3$ .

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y^{2} + 3 y + 3 y + 9 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y^{2} + 3 y + 3 y + 9 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.1.6.2

Add $3 y$ and $3 y$ .

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y^{2} + 6 y + 9 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y^{2} + 6 y + 9 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y^{2} + 6 y + 9 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.2

Simplify by adding terms.

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Step 2.1.2.1.2.1

Combine the opposite terms in $18 - y^{2} - 6 \sqrt{(3 + y) (3 - y)} + y^{2} + 6 y + 9$ .

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Step 2.1.2.1.2.1.1

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

$18 + 0 - 6 \sqrt{(3 + y) (3 - y)} + 6 y + 9 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.2.1.2

Add $18$ and $0$ .

$18 - 6 \sqrt{(3 + y) (3 - y)} + 6 y + 9 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$18 - 6 \sqrt{(3 + y) (3 - y)} + 6 y + 9 = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.1.2.1.2.2

Add $18$ and $9$ .

$27 - 6 \sqrt{(3 + y) (3 - y)} + 6 y = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$27 - 6 \sqrt{(3 + y) (3 - y)} + 6 y = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$27 - 6 \sqrt{(3 + y) (3 - y)} + 6 y = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$27 - 6 \sqrt{(3 + y) (3 - y)} + 6 y = 9$

$x = \sqrt{(3 + y) (3 - y)}$

$27 - 6 \sqrt{(3 + y) (3 - y)} + 6 y = 9$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2

Solve for $y$ in $27 - 6 \sqrt{(3 + y) (3 - y)} + 6 y = 9$ .

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Step 2.2.1

Move all terms not containing $- 6 \sqrt{(3 + y) (3 - y)}$ to the right side of the equation.

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Step 2.2.1.1

Subtract $27$ from both sides of the equation.

$- 6 \sqrt{(3 + y) (3 - y)} + 6 y = 9 - 27$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.1.2

Subtract $6 y$ from both sides of the equation.

$- 6 \sqrt{(3 + y) (3 - y)} = 9 - 27 - 6 y$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.1.3

Subtract $27$ from $9$ .

$- 6 \sqrt{(3 + y) (3 - y)} = - 18 - 6 y$

$x = \sqrt{(3 + y) (3 - y)}$

$- 6 \sqrt{(3 + y) (3 - y)} = - 18 - 6 y$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 6 \sqrt{(3 + y) (3 - y)})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3

Simplify each side of the equation.

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Step 2.2.3.1

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

${(- 6 {((3 + y) (3 - y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Simplify ${(- 6 {((3 + y) (3 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.3.2.1.1

Expand $(3 + y) (3 - y)$ using the FOIL Method.

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Step 2.2.3.2.1.1.1

Apply the distributive property.

${(- 6 {(3 (3 - y) + y (3 - y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.1.2

Apply the distributive property.

${(- 6 {(3 \cdot 3 + 3 (- y) + y (3 - y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.1.3

Apply the distributive property.

${(- 6 {(3 \cdot 3 + 3 (- y) + y \cdot 3 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

${(- 6 {(3 \cdot 3 + 3 (- y) + y \cdot 3 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.2

Simplify and combine like terms.

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Step 2.2.3.2.1.2.1

Simplify each term.

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Step 2.2.3.2.1.2.1.1

Multiply $3$ by $3$ .

${(- 6 {(9 + 3 (- y) + y \cdot 3 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.2.1.2

Multiply $- 1$ by $3$ .

${(- 6 {(9 - 3 y + y \cdot 3 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.2.1.3

Move $3$ to the left of $y$ .

${(- 6 {(9 - 3 y + 3 \cdot y + y (- y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.2.1.4

Rewrite using the commutative property of multiplication.

${(- 6 {(9 - 3 y + 3 y - y \cdot y)}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.2.1.5

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

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Step 2.2.3.2.1.2.1.5.1

Move $y$ .

${(- 6 {(9 - 3 y + 3 y - (y \cdot y))}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.2.1.5.2

Multiply $y$ by $y$ .

${(- 6 {(9 - 3 y + 3 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

${(- 6 {(9 - 3 y + 3 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

${(- 6 {(9 - 3 y + 3 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.2.2

Add $- 3 y$ and $3 y$ .

${(- 6 {(9 + 0 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.2.3

Add $9$ and $0$ .

${(- 6 {(9 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

${(- 6 {(9 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.3

Apply the product rule to $- 6 {(9 - y^{2})}^{\frac{1}{2}}$ .

${(- 6)}^{2} {({(9 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.4

Raise $- 6$ to the power of $2$ .

$36 {({(9 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.5

Multiply the exponents in ${({(9 - y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.3.2.1.5.1

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

$36 {(9 - y^{2})}^{\frac{1}{2} \cdot 2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.5.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.1.5.2.1

Cancel the common factor.

$36 {(9 - y^{2})}^{\frac{1}{2} \cdot 2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.5.2.2

Rewrite the expression.

$36 (9 - y^{2}) = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$36 (9 - y^{2}) = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$36 (9 - y^{2}) = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.6

Simplify.

$36 (9 - y^{2}) = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.7

Apply the distributive property.

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

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.8

Multiply.

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Step 2.2.3.2.1.8.1

Multiply $36$ by $9$ .

$324 + 36 (- y^{2}) = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.2.1.8.2

Multiply $- 1$ by $36$ .

$324 - 36 y^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = {(- 18 - 6 y)}^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Simplify ${(- 18 - 6 y)}^{2}$ .

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Step 2.2.3.3.1.1

Rewrite ${(- 18 - 6 y)}^{2}$ as $(- 18 - 6 y) (- 18 - 6 y)$ .

$324 - 36 y^{2} = (- 18 - 6 y) (- 18 - 6 y)$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.2

Expand $(- 18 - 6 y) (- 18 - 6 y)$ using the FOIL Method.

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Step 2.2.3.3.1.2.1

Apply the distributive property.

$324 - 36 y^{2} = - 18 (- 18 - 6 y) - 6 y (- 18 - 6 y)$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.2.2

Apply the distributive property.

$324 - 36 y^{2} = - 18 \cdot - 18 - 18 (- 6 y) - 6 y (- 18 - 6 y)$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.2.3

Apply the distributive property.

$324 - 36 y^{2} = - 18 \cdot - 18 - 18 (- 6 y) - 6 y \cdot - 18 - 6 y (- 6 y)$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = - 18 \cdot - 18 - 18 (- 6 y) - 6 y \cdot - 18 - 6 y (- 6 y)$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.3

Simplify and combine like terms.

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Step 2.2.3.3.1.3.1

Simplify each term.

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Step 2.2.3.3.1.3.1.1

Multiply $- 18$ by $- 18$ .

$324 - 36 y^{2} = 324 - 18 (- 6 y) - 6 y \cdot - 18 - 6 y (- 6 y)$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.3.1.2

Multiply $- 6$ by $- 18$ .

$324 - 36 y^{2} = 324 + 108 y - 6 y \cdot - 18 - 6 y (- 6 y)$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.3.1.3

Multiply $- 18$ by $- 6$ .

$324 - 36 y^{2} = 324 + 108 y + 108 y - 6 y (- 6 y)$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$324 - 36 y^{2} = 324 + 108 y + 108 y - 6 \cdot (- 6 y \cdot y)$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.3.1.5

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

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Step 2.2.3.3.1.3.1.5.1

Move $y$ .

$324 - 36 y^{2} = 324 + 108 y + 108 y - 6 \cdot (- 6 (y \cdot y))$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.3.1.5.2

Multiply $y$ by $y$ .

$324 - 36 y^{2} = 324 + 108 y + 108 y - 6 \cdot (- 6 y^{2})$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = 324 + 108 y + 108 y - 6 \cdot (- 6 y^{2})$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.3.1.6

Multiply $- 6$ by $- 6$ .

$324 - 36 y^{2} = 324 + 108 y + 108 y + 36 y^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = 324 + 108 y + 108 y + 36 y^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.3.3.1.3.2

Add $108 y$ and $108 y$ .

$324 - 36 y^{2} = 324 + 216 y + 36 y^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = 324 + 216 y + 36 y^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = 324 + 216 y + 36 y^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = 324 + 216 y + 36 y^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

$324 - 36 y^{2} = 324 + 216 y + 36 y^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4

Solve for $y$ .

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Step 2.2.4.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$324 + 216 y + 36 y^{2} = 324 - 36 y^{2}$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.2

Move all terms containing $y$ to the left side of the equation.

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Step 2.2.4.2.1

Add $36 y^{2}$ to both sides of the equation.

$324 + 216 y + 36 y^{2} + 36 y^{2} = 324$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.2.2

Add $36 y^{2}$ and $36 y^{2}$ .

$324 + 216 y + 72 y^{2} = 324$

$x = \sqrt{(3 + y) (3 - y)}$

$324 + 216 y + 72 y^{2} = 324$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.3

Subtract $324$ from both sides of the equation.

$324 + 216 y + 72 y^{2} - 324 = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.4

Combine the opposite terms in $324 + 216 y + 72 y^{2} - 324$ .

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Step 2.2.4.4.1

Subtract $324$ from $324$ .

$216 y + 72 y^{2} + 0 = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.4.2

Add $216 y + 72 y^{2}$ and $0$ .

$216 y + 72 y^{2} = 0$

$x = \sqrt{(3 + y) (3 - y)}$

$216 y + 72 y^{2} = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.5

Factor $72 y$ out of $216 y + 72 y^{2}$ .

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Step 2.2.4.5.1

Factor $72 y$ out of $216 y$ .

$72 y (3) + 72 y^{2} = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.5.2

Factor $72 y$ out of $72 y^{2}$ .

$72 y (3) + 72 y (y) = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.5.3

Factor $72 y$ out of $72 y (3) + 72 y (y)$ .

$72 y (3 + y) = 0$

$x = \sqrt{(3 + y) (3 - y)}$

$72 y (3 + y) = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.6

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

$y = 0$

$3 + y = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.7

Set $y$ equal to $0$ .

$y = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.8

Set $3 + y$ equal to $0$ and solve for $y$ .

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Step 2.2.4.8.1

Set $3 + y$ equal to $0$ .

$3 + y = 0$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.8.2

Subtract $3$ from both sides of the equation.

$y = - 3$

$x = \sqrt{(3 + y) (3 - y)}$

$y = - 3$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.2.4.9

The final solution is all the values that make $72 y (3 + y) = 0$ true.

$y = 0, - 3$

$x = \sqrt{(3 + y) (3 - y)}$

$y = 0, - 3$

$x = \sqrt{(3 + y) (3 - y)}$

$y = 0, - 3$

$x = \sqrt{(3 + y) (3 - y)}$

Step 2.3

Replace all occurrences of $y$ with $0$ in each equation.

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Step 2.3.1

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

$x = \sqrt{(3 + 0) (3 - (0))}$

$y = 0$

Step 2.3.2

Simplify $x = \sqrt{(3 + 0) (3 - (0))}$ .

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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.

$x = \sqrt{(3 + 0) (3 - (0))}$

$y = 0$

$x = \sqrt{(3 + 0) (3 - (0))}$

$y = 0$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $\sqrt{(3 + 0) (3 - (0))}$ .

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Step 2.3.2.2.1.1

Add $3$ and $0$ .

$x = \sqrt{3 (3 - (0))}$

$y = 0$

Step 2.3.2.2.1.2

Multiply $- 1$ by $0$ .

$x = \sqrt{3 (3 + 0)}$

$y = 0$

Step 2.3.2.2.1.3

Add $3$ and $0$ .

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

$y = 0$

Step 2.3.2.2.1.4

Multiply $3$ by $3$ .

$x = \sqrt{9}$

$y = 0$

Step 2.3.2.2.1.5

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

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

$y = 0$

Step 2.3.2.2.1.6

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

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

$x = 3$

$y = 0$

Step 2.4

Replace all occurrences of $y$ with $- 3$ in each equation.

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Step 2.4.1

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

$x = \sqrt{(3 - 3) (3 - (- 3))}$

$y = - 3$

Step 2.4.2

Simplify $x = \sqrt{(3 - 3) (3 - (- 3))}$ .

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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.

$x = \sqrt{(3 - 3) (3 - (- 3))}$

$y = - 3$

$x = \sqrt{(3 - 3) (3 - (- 3))}$

$y = - 3$

Step 2.4.2.2

Simplify the right side.

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Step 2.4.2.2.1

Simplify $\sqrt{(3 - 3) (3 - (- 3))}$ .

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Step 2.4.2.2.1.1

Subtract $3$ from $3$ .

$x = \sqrt{0 (3 - (- 3))}$

$y = - 3$

Step 2.4.2.2.1.2

Multiply $- 1$ by $- 3$ .

$x = \sqrt{0 (3 + 3)}$

$y = - 3$

Step 2.4.2.2.1.3

Add $3$ and $3$ .

$x = \sqrt{0 \cdot 6}$

$y = - 3$

Step 2.4.2.2.1.4

Multiply $0$ by $6$ .

$x = \sqrt{0}$

$y = - 3$

Step 2.4.2.2.1.5

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

$x = \sqrt{0^{2}}$

$y = - 3$

Step 2.4.2.2.1.6

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

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

Step 3

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

$(3, 0)$

$(0, - 3)$

Step 4

The result can be shown in multiple forms.

Point Form:

$(3, 0), (0, - 3), (0, - 3)$

Equation Form:

$x = 3, y = 0$

$x = 0, y = - 3$

Step 5