Find the Expanded Form

$\frac{{(x - 2)}^{2}}{2} + \frac{{(y + 5)}^{2}}{4} = 12$

Step 1

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

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Step 1.1

To write $\frac{{(x - 2)}^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{{(x - 2)}^{2}}{2} \cdot \frac{2}{2} + \frac{{(y + 5)}^{2}}{4} = 12$

Step 1.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{{(x - 2)}^{2}}{2}$ by $\frac{2}{2}$ .

$\frac{{(x - 2)}^{2} \cdot 2}{2 \cdot 2} + \frac{{(y + 5)}^{2}}{4} = 12$

Step 1.2.2

Multiply $2$ by $2$ .

$\frac{{(x - 2)}^{2} \cdot 2}{4} + \frac{{(y + 5)}^{2}}{4} = 12$

Step 1.3

Combine the numerators over the common denominator.

$\frac{{(x - 2)}^{2} \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

$\frac{(x - 2) (x - 2) \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.4.2.1

Apply the distributive property.

$\frac{(x (x - 2) - 2 (x - 2)) \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.2.2

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 2 - 2 (x - 2)) \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.2.3

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.3

Simplify and combine like terms.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Multiply $x$ by $x$ .

$\frac{(x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.3.1.2

Move $- 2$ to the left of $x$ .

$\frac{(x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.3.1.3

Multiply $- 2$ by $- 2$ .

$\frac{(x^{2} - 2 x - 2 x + 4) \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.3.2

Subtract $2 x$ from $- 2 x$ .

$\frac{(x^{2} - 4 x + 4) \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.4

Apply the distributive property.

$\frac{x^{2} \cdot 2 - 4 x \cdot 2 + 4 \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.5

Simplify.

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Step 1.4.5.1

Move $2$ to the left of $x^{2}$ .

$\frac{2 \cdot x^{2} - 4 x \cdot 2 + 4 \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.5.2

Multiply $2$ by $- 4$ .

$\frac{2 \cdot x^{2} - 8 x + 4 \cdot 2 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.5.3

Multiply $4$ by $2$ .

$\frac{2 \cdot x^{2} - 8 x + 8 + {(y + 5)}^{2}}{4} = 12$

Step 1.4.6

Rewrite ${(y + 5)}^{2}$ as $(y + 5) (y + 5)$ .

$\frac{2 x^{2} - 8 x + 8 + (y + 5) (y + 5)}{4} = 12$

Step 1.4.7

Expand $(y + 5) (y + 5)$ using the FOIL Method.

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Step 1.4.7.1

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 8 + y (y + 5) + 5 (y + 5)}{4} = 12$

Step 1.4.7.2

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 8 + y \cdot y + y \cdot 5 + 5 (y + 5)}{4} = 12$

Step 1.4.7.3

Apply the distributive property.

$\frac{2 x^{2} - 8 x + 8 + y \cdot y + y \cdot 5 + 5 y + 5 \cdot 5}{4} = 12$

Step 1.4.8

Simplify and combine like terms.

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Step 1.4.8.1

Simplify each term.

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Step 1.4.8.1.1

Multiply $y$ by $y$ .

$\frac{2 x^{2} - 8 x + 8 + y^{2} + y \cdot 5 + 5 y + 5 \cdot 5}{4} = 12$

Step 1.4.8.1.2

Move $5$ to the left of $y$ .

$\frac{2 x^{2} - 8 x + 8 + y^{2} + 5 \cdot y + 5 y + 5 \cdot 5}{4} = 12$

Step 1.4.8.1.3

Multiply $5$ by $5$ .

$\frac{2 x^{2} - 8 x + 8 + y^{2} + 5 y + 5 y + 25}{4} = 12$

Step 1.4.8.2

Add $5 y$ and $5 y$ .

$\frac{2 x^{2} - 8 x + 8 + y^{2} + 10 y + 25}{4} = 12$

Step 1.4.9

Add $8$ and $25$ .

$\frac{2 x^{2} - 8 x + y^{2} + 10 y + 33}{4} = 12$

Step 2

Multiply both sides by $4$ .

$\frac{2 x^{2} - 8 x + y^{2} + 10 y + 33}{4} \cdot 4 = 12 \cdot 4$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{2 x^{2} - 8 x + y^{2} + 10 y + 33}{4} \cdot 4$ .

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Step 3.1.1.1

Cancel the common factor of $4$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{2 x^{2} - 8 x + y^{2} + 10 y + 33}{4} \cdot 4 = 12 \cdot 4$

Step 3.1.1.1.2

Rewrite the expression.

$2 x^{2} - 8 x + y^{2} + 10 y + 33 = 12 \cdot 4$

Step 3.1.1.2

Move $- 8 x$ .

$2 x^{2} + y^{2} - 8 x + 10 y + 33 = 12 \cdot 4$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $12$ by $4$ .

$2 x^{2} + y^{2} - 8 x + 10 y + 33 = 48$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract $48$ from both sides of the equation.

$2 x^{2} + y^{2} - 8 x + 10 y + 33 - 48 = 0$

Step 4.2

Subtract $48$ from $33$ .

$2 x^{2} + y^{2} - 8 x + 10 y - 15 = 0$

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