Examples

{(x + 3)}^{2} + {(y - 6)}^{2} = 1

${(x + 3)}^{2} + {(y - 6)}^{2} = 1$

Step 1

Simplify the left side

{(x + 3)}^{2} + {(y - 6)}^{2}

${(x + 3)}^{2} + {(y - 6)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

{(x + 3)}^{2}

${(x + 3)}^{2}$ as

(x + 3) (x + 3)

$(x + 3) (x + 3)$ .

(x + 3) (x + 3) + {(y - 6)}^{2} = 1

$(x + 3) (x + 3) + {(y - 6)}^{2} = 1$

Step 1.1.2

Expand

(x + 3) (x + 3)

$(x + 3) (x + 3)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

x (x + 3) + 3 (x + 3) + {(y - 6)}^{2} = 1

$x (x + 3) + 3 (x + 3) + {(y - 6)}^{2} = 1$

Step 1.1.2.2

Apply the distributive property.

x \cdot x + x \cdot 3 + 3 (x + 3) + {(y - 6)}^{2} = 1

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

Step 1.1.2.3

Apply the distributive property.

x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + {(y - 6)}^{2} = 1

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

x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + {(y - 6)}^{2} = 1

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

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply

x

$x$ by

x

$x$ .

x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 + {(y - 6)}^{2} = 1

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

Step 1.1.3.1.2

Move

3

$3$ to the left of

x

$x$ .

x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 + {(y - 6)}^{2} = 1

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

Step 1.1.3.1.3

Multiply

3

$3$ by

3

$3$ .

x^{2} + 3 x + 3 x + 9 + {(y - 6)}^{2} = 1

$x^{2} + 3 x + 3 x + 9 + {(y - 6)}^{2} = 1$

x^{2} + 3 x + 3 x + 9 + {(y - 6)}^{2} = 1

$x^{2} + 3 x + 3 x + 9 + {(y - 6)}^{2} = 1$

Step 1.1.3.2

Add

3 x

$3 x$ and

3 x

$3 x$ .

x^{2} + 6 x + 9 + {(y - 6)}^{2} = 1

$x^{2} + 6 x + 9 + {(y - 6)}^{2} = 1$

x^{2} + 6 x + 9 + {(y - 6)}^{2} = 1

$x^{2} + 6 x + 9 + {(y - 6)}^{2} = 1$

Step 1.1.4

Rewrite

{(y - 6)}^{2}

${(y - 6)}^{2}$ as

(y - 6) (y - 6)

$(y - 6) (y - 6)$ .

x^{2} + 6 x + 9 + (y - 6) (y - 6) = 1

$x^{2} + 6 x + 9 + (y - 6) (y - 6) = 1$

Step 1.1.5

Expand

(y - 6) (y - 6)

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

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Step 1.1.5.1

Apply the distributive property.

x^{2} + 6 x + 9 + y (y - 6) - 6 (y - 6) = 1

$x^{2} + 6 x + 9 + y (y - 6) - 6 (y - 6) = 1$

Step 1.1.5.2

Apply the distributive property.

x^{2} + 6 x + 9 + y \cdot y + y \cdot - 6 - 6 (y - 6) = 1

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

Step 1.1.5.3

Apply the distributive property.

x^{2} + 6 x + 9 + y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6 = 1

$x^{2} + 6 x + 9 + y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6 = 1$

x^{2} + 6 x + 9 + y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6 = 1

$x^{2} + 6 x + 9 + y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6 = 1$

Step 1.1.6

Simplify and combine like terms.

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Step 1.1.6.1

Simplify each term.

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Step 1.1.6.1.1

Multiply

y

$y$ by

y

$y$ .

x^{2} + 6 x + 9 + y^{2} + y \cdot - 6 - 6 y - 6 \cdot - 6 = 1

$x^{2} + 6 x + 9 + y^{2} + y \cdot - 6 - 6 y - 6 \cdot - 6 = 1$

Step 1.1.6.1.2

Move

- 6

$- 6$ to the left of

y

$y$ .

x^{2} + 6 x + 9 + y^{2} - 6 \cdot y - 6 y - 6 \cdot - 6 = 1

$x^{2} + 6 x + 9 + y^{2} - 6 \cdot y - 6 y - 6 \cdot - 6 = 1$

Step 1.1.6.1.3

Multiply

- 6

$- 6$ by

- 6

$- 6$ .

x^{2} + 6 x + 9 + y^{2} - 6 y - 6 y + 36 = 1

$x^{2} + 6 x + 9 + y^{2} - 6 y - 6 y + 36 = 1$

x^{2} + 6 x + 9 + y^{2} - 6 y - 6 y + 36 = 1

$x^{2} + 6 x + 9 + y^{2} - 6 y - 6 y + 36 = 1$

Step 1.1.6.2

Subtract

6 y

$6 y$ from

- 6 y

$- 6 y$ .

x^{2} + 6 x + 9 + y^{2} - 12 y + 36 = 1

$x^{2} + 6 x + 9 + y^{2} - 12 y + 36 = 1$

x^{2} + 6 x + 9 + y^{2} - 12 y + 36 = 1

$x^{2} + 6 x + 9 + y^{2} - 12 y + 36 = 1$

x^{2} + 6 x + 9 + y^{2} - 12 y + 36 = 1

$x^{2} + 6 x + 9 + y^{2} - 12 y + 36 = 1$

Step 1.2

Simplify the expression.

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Step 1.2.1

Add

9

$9$ and

36

$36$ .

x^{2} + 6 x + y^{2} - 12 y + 45 = 1

$x^{2} + 6 x + y^{2} - 12 y + 45 = 1$

Step 1.2.2

Move

6 x

$6 x$ .

x^{2} + y^{2} + 6 x - 12 y + 45 = 1

$x^{2} + y^{2} + 6 x - 12 y + 45 = 1$

x^{2} + y^{2} + 6 x - 12 y + 45 = 1

$x^{2} + y^{2} + 6 x - 12 y + 45 = 1$

x^{2} + y^{2} + 6 x - 12 y + 45 = 1

$x^{2} + y^{2} + 6 x - 12 y + 45 = 1$

Step 2

Subtract

1

$1$ from both sides of the equation.

x^{2} + y^{2} + 6 x - 12 y + 45 - 1 = 0

$x^{2} + y^{2} + 6 x - 12 y + 45 - 1 = 0$

Step 3

Subtract

1

$1$ from

45

$45$ .

x^{2} + y^{2} + 6 x - 12 y + 44 = 0

$x^{2} + y^{2} + 6 x - 12 y + 44 = 0$

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