Precalculus Examples

${(2 x + 3)}^{2} + {(2 y - 6)}^{2} = 4$

Simplify the left side ${(2 x + 3)}^{2} + {(2 y - 6)}^{2}$ .

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Simplify each term.

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

$(2 x + 3) (2 x + 3) + {(2 y - 6)}^{2} = 4$

Expand $(2 x + 3) (2 x + 3)$ using the FOIL Method.

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Apply the distributive property.

$2 x (2 x + 3) + 3 (2 x + 3) + {(2 y - 6)}^{2} = 4$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 3 + 3 (2 x + 3) + {(2 y - 6)}^{2} = 4$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + {(2 y - 6)}^{2} = 4$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$2 \cdot (2 x^{2}) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + {(2 y - 6)}^{2} = 4$

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 + {(2 y - 6)}^{2} = 4$

Multiply $3$ by $2$ .

$4 x^{2} + 6 x + 3 (2 x) + 3 \cdot 3 + {(2 y - 6)}^{2} = 4$

Multiply $2$ by $3$ .

$4 x^{2} + 6 x + 6 x + 3 \cdot 3 + {(2 y - 6)}^{2} = 4$

Multiply $3$ by $3$ .

$4 x^{2} + 6 x + 6 x + 9 + {(2 y - 6)}^{2} = 4$

Add $6 x$ and $6 x$ .

$4 x^{2} + 12 x + 9 + {(2 y - 6)}^{2} = 4$

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

$4 x^{2} + 12 x + 9 + (2 y - 6) (2 y - 6) = 4$

Expand $(2 y - 6) (2 y - 6)$ using the FOIL Method.

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Apply the distributive property.

$4 x^{2} + 12 x + 9 + 2 y (2 y - 6) - 6 (2 y - 6) = 4$

Apply the distributive property.

$4 x^{2} + 12 x + 9 + 2 y (2 y) + 2 y \cdot - 6 - 6 (2 y - 6) = 4$

Apply the distributive property.

$4 x^{2} + 12 x + 9 + 2 y (2 y) + 2 y \cdot - 6 - 6 (2 y) - 6 \cdot - 6 = 4$

Simplify and combine like terms.

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Simplify each term.

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Multiply $y$ by $y$ .

$4 x^{2} + 12 x + 9 + 2 \cdot (2 y^{2}) + 2 y \cdot - 6 - 6 (2 y) - 6 \cdot - 6 = 4$

Multiply $2$ by $2$ .

$4 x^{2} + 12 x + 9 + 4 y^{2} + 2 y \cdot - 6 - 6 (2 y) - 6 \cdot - 6 = 4$

Multiply $- 6$ by $2$ .

$4 x^{2} + 12 x + 9 + 4 y^{2} - 12 y - 6 (2 y) - 6 \cdot - 6 = 4$

Multiply $2$ by $- 6$ .

$4 x^{2} + 12 x + 9 + 4 y^{2} - 12 y - 12 y - 6 \cdot - 6 = 4$

Multiply $- 6$ by $- 6$ .

$4 x^{2} + 12 x + 9 + 4 y^{2} - 12 y - 12 y + 36 = 4$

Subtract $12 y$ from $- 12 y$ .

$4 x^{2} + 12 x + 9 + 4 y^{2} - 24 y + 36 = 4$

Simplify the expression.

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Add $9$ and $36$ .

$4 x^{2} + 12 x + 4 y^{2} - 24 y + 45 = 4$

Move $12 x$ .

$4 x^{2} + 4 y^{2} + 12 x - 24 y + 45 = 4$

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

$4 x^{2} + 4 y^{2} + 12 x - 24 y + 45 - 4 = 0$

Subtract $4$ from $45$ .

$4 x^{2} + 4 y^{2} + 12 x - 24 y + 41 = 0$

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