Precalculus Examples

${(x - 5)}^{2} + {(y - 4)}^{2} = 4$

Step 1

Simplify the left side ${(x - 5)}^{2} + {(y - 4)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$(x - 5) (x - 5) + {(y - 4)}^{2} = 4$

Step 1.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$x (x - 5) - 5 (x - 5) + {(y - 4)}^{2} = 4$

Step 1.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 5 (x - 5) + {(y - 4)}^{2} = 4$

Step 1.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 + {(y - 4)}^{2} = 4$

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 $x$ .

$x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 + {(y - 4)}^{2} = 4$

Step 1.1.3.1.2

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 + {(y - 4)}^{2} = 4$

Step 1.1.3.1.3

Multiply $- 5$ by $- 5$ .

$x^{2} - 5 x - 5 x + 25 + {(y - 4)}^{2} = 4$

Step 1.1.3.2

Subtract $5 x$ from $- 5 x$ .

$x^{2} - 10 x + 25 + {(y - 4)}^{2} = 4$

Step 1.1.4

Rewrite ${(y - 4)}^{2}$ as $(y - 4) (y - 4)$ .

$x^{2} - 10 x + 25 + (y - 4) (y - 4) = 4$

Step 1.1.5

Expand $(y - 4) (y - 4)$ using the FOIL Method.

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Step 1.1.5.1

Apply the distributive property.

$x^{2} - 10 x + 25 + y (y - 4) - 4 (y - 4) = 4$

Step 1.1.5.2

Apply the distributive property.

$x^{2} - 10 x + 25 + y \cdot y + y \cdot - 4 - 4 (y - 4) = 4$

Step 1.1.5.3

Apply the distributive property.

$x^{2} - 10 x + 25 + y \cdot y + y \cdot - 4 - 4 y - 4 \cdot - 4 = 4$

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 $y$ .

$x^{2} - 10 x + 25 + y^{2} + y \cdot - 4 - 4 y - 4 \cdot - 4 = 4$

Step 1.1.6.1.2

Move $- 4$ to the left of $y$ .

$x^{2} - 10 x + 25 + y^{2} - 4 \cdot y - 4 y - 4 \cdot - 4 = 4$

Step 1.1.6.1.3

Multiply $- 4$ by $- 4$ .

$x^{2} - 10 x + 25 + y^{2} - 4 y - 4 y + 16 = 4$

Step 1.1.6.2

Subtract $4 y$ from $- 4 y$ .

$x^{2} - 10 x + 25 + y^{2} - 8 y + 16 = 4$

Step 1.2

Simplify the expression.

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Step 1.2.1

Add $25$ and $16$ .

$x^{2} - 10 x + y^{2} - 8 y + 41 = 4$

Step 1.2.2

Move $- 10 x$ .

$x^{2} + y^{2} - 10 x - 8 y + 41 = 4$

Step 2

Subtract $4$ from both sides of the equation.

$x^{2} + y^{2} - 10 x - 8 y + 41 - 4 = 0$

Step 3

Subtract $4$ from $41$ .

$x^{2} + y^{2} - 10 x - 8 y + 37 = 0$

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