Precalculus Examples

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

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

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Simplify each term.

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Remove parentheses.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ by adding the exponents.

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Combine $x$ and $x$

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Raise $x$ to the power of $1$ .

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

Raise $x$ to the power of $1$ .

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

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

$x^{1 + 1} + x \cdot - 5 - 5 x - 5 \cdot - 5 + {(y - 4)}^{2} = 4$

Add $1$ and $1$ .

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

Move $- 5$ to the left of the expression $x \cdot - 5$ .

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

Multiply $- 5$ by $x$ .

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

Multiply $- 5$ by $- 5$ .

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

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

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

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Remove parentheses.

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

Simplify and combine like terms.

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Simplify each term.

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

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Combine $y$ and $y$

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Raise $y$ to the power of $1$ .

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

Raise $y$ to the power of $1$ .

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

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

$x^{2} - 10 x + 25 + (y^{1 + 1} + y \cdot - 4 - 4 y - 4 \cdot - 4) = 4$

Add $1$ and $1$ .

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

Move $- 4$ to the left of the expression $y \cdot - 4$ .

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

Multiply $- 4$ by $y$ .

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

Multiply $- 4$ by $- 4$ .

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

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

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

Simplify the expression.

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Remove unnecessary parentheses.

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

Add $25$ and $16$ .

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

Move $- 10 x$ .

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

Move $4$ to the left side of the equation by subtracting it from both sides.

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

Subtract $4$ from $41$ .

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

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