Precalculus Examples

{(x - 8)}^{2} + {(y + 2)}^{2} = 1

${(x - 8)}^{2} + {(y + 2)}^{2} = 1$

Step 1

Simplify the left side

{(x - 8)}^{2} + {(y + 2)}^{2}

${(x - 8)}^{2} + {(y + 2)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

{(x - 8)}^{2}

${(x - 8)}^{2}$ as

(x - 8) (x - 8)

$(x - 8) (x - 8)$ .

(x - 8) (x - 8) + {(y + 2)}^{2} = 1

$(x - 8) (x - 8) + {(y + 2)}^{2} = 1$

Step 1.1.2

Expand

(x - 8) (x - 8)

$(x - 8) (x - 8)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

x (x - 8) - 8 (x - 8) + {(y + 2)}^{2} = 1

$x (x - 8) - 8 (x - 8) + {(y + 2)}^{2} = 1$

Step 1.1.2.2

Apply the distributive property.

x \cdot x + x \cdot - 8 - 8 (x - 8) + {(y + 2)}^{2} = 1

$x \cdot x + x \cdot - 8 - 8 (x - 8) + {(y + 2)}^{2} = 1$

Step 1.1.2.3

Apply the distributive property.

x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 + {(y + 2)}^{2} = 1

$x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 + {(y + 2)}^{2} = 1$

x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 + {(y + 2)}^{2} = 1

$x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 + {(y + 2)}^{2} = 1$

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 - 8 - 8 x - 8 \cdot - 8 + {(y + 2)}^{2} = 1

$x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8 + {(y + 2)}^{2} = 1$

Step 1.1.3.1.2

Move

- 8

$- 8$ to the left of

x

$x$ .

x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8 + {(y + 2)}^{2} = 1

$x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8 + {(y + 2)}^{2} = 1$

Step 1.1.3.1.3

Multiply

- 8

$- 8$ by

- 8

$- 8$ .

x^{2} - 8 x - 8 x + 64 + {(y + 2)}^{2} = 1

$x^{2} - 8 x - 8 x + 64 + {(y + 2)}^{2} = 1$

x^{2} - 8 x - 8 x + 64 + {(y + 2)}^{2} = 1

$x^{2} - 8 x - 8 x + 64 + {(y + 2)}^{2} = 1$

Step 1.1.3.2

Subtract

8 x

$8 x$ from

- 8 x

$- 8 x$ .

x^{2} - 16 x + 64 + {(y + 2)}^{2} = 1

$x^{2} - 16 x + 64 + {(y + 2)}^{2} = 1$

x^{2} - 16 x + 64 + {(y + 2)}^{2} = 1

$x^{2} - 16 x + 64 + {(y + 2)}^{2} = 1$

Step 1.1.4

Rewrite

{(y + 2)}^{2}

${(y + 2)}^{2}$ as

(y + 2) (y + 2)

$(y + 2) (y + 2)$ .

x^{2} - 16 x + 64 + (y + 2) (y + 2) = 1

$x^{2} - 16 x + 64 + (y + 2) (y + 2) = 1$

Step 1.1.5

Expand

(y + 2) (y + 2)

$(y + 2) (y + 2)$ using the FOIL Method.

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Step 1.1.5.1

Apply the distributive property.

x^{2} - 16 x + 64 + y (y + 2) + 2 (y + 2) = 1

$x^{2} - 16 x + 64 + y (y + 2) + 2 (y + 2) = 1$

Step 1.1.5.2

Apply the distributive property.

x^{2} - 16 x + 64 + y \cdot y + y \cdot 2 + 2 (y + 2) = 1

$x^{2} - 16 x + 64 + y \cdot y + y \cdot 2 + 2 (y + 2) = 1$

Step 1.1.5.3

Apply the distributive property.

x^{2} - 16 x + 64 + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2 = 1

$x^{2} - 16 x + 64 + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2 = 1$

x^{2} - 16 x + 64 + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2 = 1

$x^{2} - 16 x + 64 + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2 = 1$

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} - 16 x + 64 + y^{2} + y \cdot 2 + 2 y + 2 \cdot 2 = 1

$x^{2} - 16 x + 64 + y^{2} + y \cdot 2 + 2 y + 2 \cdot 2 = 1$

Step 1.1.6.1.2

Move

2

$2$ to the left of

y

$y$ .

x^{2} - 16 x + 64 + y^{2} + 2 \cdot y + 2 y + 2 \cdot 2 = 1

$x^{2} - 16 x + 64 + y^{2} + 2 \cdot y + 2 y + 2 \cdot 2 = 1$

Step 1.1.6.1.3

Multiply

2

$2$ by

2

$2$ .

x^{2} - 16 x + 64 + y^{2} + 2 y + 2 y + 4 = 1

$x^{2} - 16 x + 64 + y^{2} + 2 y + 2 y + 4 = 1$

x^{2} - 16 x + 64 + y^{2} + 2 y + 2 y + 4 = 1

$x^{2} - 16 x + 64 + y^{2} + 2 y + 2 y + 4 = 1$

Step 1.1.6.2

Add

2 y

$2 y$ and

2 y

$2 y$ .

x^{2} - 16 x + 64 + y^{2} + 4 y + 4 = 1

$x^{2} - 16 x + 64 + y^{2} + 4 y + 4 = 1$

x^{2} - 16 x + 64 + y^{2} + 4 y + 4 = 1

$x^{2} - 16 x + 64 + y^{2} + 4 y + 4 = 1$

x^{2} - 16 x + 64 + y^{2} + 4 y + 4 = 1

$x^{2} - 16 x + 64 + y^{2} + 4 y + 4 = 1$

Step 1.2

Simplify the expression.

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Step 1.2.1

Add

64

$64$ and

4

$4$ .

x^{2} - 16 x + y^{2} + 4 y + 68 = 1

$x^{2} - 16 x + y^{2} + 4 y + 68 = 1$

Step 1.2.2

Move

- 16 x

$- 16 x$ .

x^{2} + y^{2} - 16 x + 4 y + 68 = 1

$x^{2} + y^{2} - 16 x + 4 y + 68 = 1$

x^{2} + y^{2} - 16 x + 4 y + 68 = 1

$x^{2} + y^{2} - 16 x + 4 y + 68 = 1$

x^{2} + y^{2} - 16 x + 4 y + 68 = 1

$x^{2} + y^{2} - 16 x + 4 y + 68 = 1$

Step 2

Subtract

1

$1$ from both sides of the equation.

x^{2} + y^{2} - 16 x + 4 y + 68 - 1 = 0

$x^{2} + y^{2} - 16 x + 4 y + 68 - 1 = 0$

Step 3

Subtract

1

$1$ from

68

$68$ .

x^{2} + y^{2} - 16 x + 4 y + 67 = 0

$x^{2} + y^{2} - 16 x + 4 y + 67 = 0$