Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

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 $y$ by $y$ .

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

Step 1.1.3.1.2

Move $4$ to the left of $y$ .

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

Step 1.1.3.1.3

Multiply $4$ by $4$ .

$\frac{{(x - 4)}^{2}}{2} + y^{2} + 4 y + 4 y + 16 = 12$

Step 1.1.3.2

Add $4 y$ and $4 y$ .

$\frac{{(x - 4)}^{2}}{2} + y^{2} + 8 y + 16 = 12$

Step 1.2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{{(x - 4)}^{2}}{2} + y^{2} \cdot \frac{2}{2} + 8 y + 16 = 12$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $y^{2}$ and $\frac{2}{2}$ .

$\frac{{(x - 4)}^{2}}{2} + \frac{y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{{(x - 4)}^{2} + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

$\frac{(x - 4) (x - 4) + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.4.2

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

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Step 1.4.2.1

Apply the distributive property.

$\frac{x (x - 4) - 4 (x - 4) + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.4.2.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 - 4 (x - 4) + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.4.2.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

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 $x$ .

$\frac{x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4 + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.4.3.1.2

Move $- 4$ to the left of $x$ .

$\frac{x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4 + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.4.3.1.3

Multiply $- 4$ by $- 4$ .

$\frac{x^{2} - 4 x - 4 x + 16 + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.4.3.2

Subtract $4 x$ from $- 4 x$ .

$\frac{x^{2} - 8 x + 16 + y^{2} \cdot 2}{2} + 8 y + 16 = 12$

Step 1.4.4

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

$\frac{x^{2} - 8 x + 16 + 2 y^{2}}{2} + 8 y + 16 = 12$

Step 1.5

To write $8 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2} - 8 x + 16 + 2 y^{2}}{2} + 8 y \cdot \frac{2}{2} + 16 = 12$

Step 1.6

Combine $8 y$ and $\frac{2}{2}$ .

$\frac{x^{2} - 8 x + 16 + 2 y^{2}}{2} + \frac{8 y \cdot 2}{2} + 16 = 12$

Step 1.7

Combine the numerators over the common denominator.

$\frac{x^{2} - 8 x + 16 + 2 y^{2} + 8 y \cdot 2}{2} + 16 = 12$

Step 1.8

Multiply $2$ by $8$ .

$\frac{x^{2} - 8 x + 16 + 2 y^{2} + 16 y}{2} + 16 = 12$

Step 1.9

To write $16$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2} - 8 x + 16 + 2 y^{2} + 16 y}{2} + 16 \cdot \frac{2}{2} = 12$

Step 1.10

Simplify terms.

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Step 1.10.1

Combine $16$ and $\frac{2}{2}$ .

$\frac{x^{2} - 8 x + 16 + 2 y^{2} + 16 y}{2} + \frac{16 \cdot 2}{2} = 12$

Step 1.10.2

Combine the numerators over the common denominator.

$\frac{x^{2} - 8 x + 16 + 2 y^{2} + 16 y + 16 \cdot 2}{2} = 12$

Step 1.11

Simplify the numerator.

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Step 1.11.1

Multiply $16$ by $2$ .

$\frac{x^{2} - 8 x + 16 + 2 y^{2} + 16 y + 32}{2} = 12$

Step 1.11.2

Add $16$ and $32$ .

$\frac{x^{2} - 8 x + 2 y^{2} + 16 y + 48}{2} = 12$

Step 2

Multiply both sides by $2$ .

$\frac{x^{2} - 8 x + 2 y^{2} + 16 y + 48}{2} \cdot 2 = 12 \cdot 2$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{x^{2} - 8 x + 2 y^{2} + 16 y + 48}{2} \cdot 2$ .

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Step 3.1.1.1

Cancel the common factor of $2$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{x^{2} - 8 x + 2 y^{2} + 16 y + 48}{2} \cdot 2 = 12 \cdot 2$

Step 3.1.1.1.2

Rewrite the expression.

$x^{2} - 8 x + 2 y^{2} + 16 y + 48 = 12 \cdot 2$

Step 3.1.1.2

Move $- 8 x$ .

$x^{2} + 2 y^{2} - 8 x + 16 y + 48 = 12 \cdot 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $12$ by $2$ .

$x^{2} + 2 y^{2} - 8 x + 16 y + 48 = 24$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract $24$ from both sides of the equation.

$x^{2} + 2 y^{2} - 8 x + 16 y + 48 - 24 = 0$

Step 4.2

Subtract $24$ from $48$ .

$x^{2} + 2 y^{2} - 8 x + 16 y + 24 = 0$

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