Precalculus Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

Move $x$ .

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

Step 1.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.1.3.1.4

Multiply $3$ by $2$ .

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

Step 1.1.3.1.5

Multiply $2$ by $3$ .

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

Step 1.1.3.1.6

Multiply $3$ by $3$ .

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

Step 1.1.3.2

Add $6 x$ and $6 x$ .

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

Step 1.1.4

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

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

Step 1.1.5

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

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Step 1.1.5.1

Apply the distributive property.

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

Step 1.1.5.2

Apply the distributive property.

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

Step 1.1.5.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 1.1.6.1.2

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

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Step 1.1.6.1.2.1

Move $y$ .

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

Step 1.1.6.1.2.2

Multiply $y$ by $y$ .

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

Step 1.1.6.1.3

Multiply $2$ by $2$ .

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

Step 1.1.6.1.4

Multiply $- 6$ by $2$ .

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

Step 1.1.6.1.5

Multiply $2$ by $- 6$ .

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

Step 1.1.6.1.6

Multiply $- 6$ by $- 6$ .

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

Step 1.1.6.2

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

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Add $9$ and $36$ .

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

Step 1.2.2

Move $12 x$ .

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

Step 2

Subtract $4$ from both sides of the equation.

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

Step 3

Subtract $4$ from $45$ .

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

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