Precalculus Examples

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

Step 1

Simplify the left side ${(x - 8)}^{2} + {(y + 2)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

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 $x$ by $x$ .

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

Step 1.1.3.1.2

Move $- 8$ to the left of $x$ .

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

Step 1.1.3.1.3

Multiply $- 8$ by $- 8$ .

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

Step 1.1.3.2

Subtract $8 x$ from $- 8 x$ .

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

Step 1.1.4

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

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

Step 1.1.5

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

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Step 1.1.5.1

Apply the distributive property.

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

Step 1.1.5.2

Apply the distributive property.

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

Step 1.1.5.3

Apply the distributive property.

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

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 $y$ by $y$ .

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

Step 1.1.6.1.2

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

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

Step 1.1.6.1.3

Multiply $2$ by $2$ .

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

Step 1.1.6.2

Add $2 y$ and $2 y$ .

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Add $64$ and $4$ .

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

Step 1.2.2

Move $- 16 x$ .

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

Step 2

Subtract $1$ from both sides of the equation.

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

Step 3

Subtract $1$ from $68$ .

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