Algebra Examples

${(x + 3)}^{2} + {(y - 6)}^{2} = 1$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$(x + 3) (x + 3) + {(y - 6)}^{2} = 1$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$x (x + 3) + 3 (x + 3) + {(y - 6)}^{2} = 1$

Step 1.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 (x + 3) + {(y - 6)}^{2} = 1$

Step 1.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + {(y - 6)}^{2} = 1$

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 $x$ .

$x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 + {(y - 6)}^{2} = 1$

Step 1.1.3.1.2

Move $3$ to the left of $x$ .

$x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 + {(y - 6)}^{2} = 1$

Step 1.1.3.1.3

Multiply $3$ by $3$ .

$x^{2} + 3 x + 3 x + 9 + {(y - 6)}^{2} = 1$

Step 1.1.3.2

Add $3 x$ and $3 x$ .

$x^{2} + 6 x + 9 + {(y - 6)}^{2} = 1$

Step 1.1.4

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

$x^{2} + 6 x + 9 + (y - 6) (y - 6) = 1$

Step 1.1.5

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

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Step 1.1.5.1

Apply the distributive property.

$x^{2} + 6 x + 9 + y (y - 6) - 6 (y - 6) = 1$

Step 1.1.5.2

Apply the distributive property.

$x^{2} + 6 x + 9 + y \cdot y + y \cdot - 6 - 6 (y - 6) = 1$

Step 1.1.5.3

Apply the distributive property.

$x^{2} + 6 x + 9 + y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6 = 1$

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 $y$ .

$x^{2} + 6 x + 9 + y^{2} + y \cdot - 6 - 6 y - 6 \cdot - 6 = 1$

Step 1.1.6.1.2

Move $- 6$ to the left of $y$ .

$x^{2} + 6 x + 9 + y^{2} - 6 \cdot y - 6 y - 6 \cdot - 6 = 1$

Step 1.1.6.1.3

Multiply $- 6$ by $- 6$ .

$x^{2} + 6 x + 9 + y^{2} - 6 y - 6 y + 36 = 1$

Step 1.1.6.2

Subtract $6 y$ from $- 6 y$ .

$x^{2} + 6 x + 9 + y^{2} - 12 y + 36 = 1$

Step 1.2

Simplify the expression.

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Step 1.2.1

Add $9$ and $36$ .

$x^{2} + 6 x + y^{2} - 12 y + 45 = 1$

Step 1.2.2

Move $6 x$ .

$x^{2} + y^{2} + 6 x - 12 y + 45 = 1$

Step 2

Subtract $1$ from both sides of the equation.

$x^{2} + y^{2} + 6 x - 12 y + 45 - 1 = 0$

Step 3

Subtract $1$ from $45$ .

$x^{2} + y^{2} + 6 x - 12 y + 44 = 0$

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