Examples

${(x + 4)}^{2} - {(y + 4)}^{2} - 4 = 0$

Step 1

Simplify each term.

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Step 1.1

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

$(x + 4) (x + 4) - {(y + 4)}^{2} - 4 = 0$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$x (x + 4) + 4 (x + 4) - {(y + 4)}^{2} - 4 = 0$

Step 1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 (x + 4) - {(y + 4)}^{2} - 4 = 0$

Step 1.2.3

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 - {(y + 4)}^{2} - 4 = 0$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 - {(y + 4)}^{2} - 4 = 0$

Step 1.3.1.2

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

$x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 - {(y + 4)}^{2} - 4 = 0$

Step 1.3.1.3

Multiply $4$ by $4$ .

$x^{2} + 4 x + 4 x + 16 - {(y + 4)}^{2} - 4 = 0$

Step 1.3.2

Add $4 x$ and $4 x$ .

$x^{2} + 8 x + 16 - {(y + 4)}^{2} - 4 = 0$

Step 1.4

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

$x^{2} + 8 x + 16 - ((y + 4) (y + 4)) - 4 = 0$

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

$x^{2} + 8 x + 16 - (y (y + 4) + 4 (y + 4)) - 4 = 0$

Step 1.5.2

Apply the distributive property.

$x^{2} + 8 x + 16 - (y \cdot y + y \cdot 4 + 4 (y + 4)) - 4 = 0$

Step 1.5.3

Apply the distributive property.

$x^{2} + 8 x + 16 - (y \cdot y + y \cdot 4 + 4 y + 4 \cdot 4) - 4 = 0$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $y$ by $y$ .

$x^{2} + 8 x + 16 - (y^{2} + y \cdot 4 + 4 y + 4 \cdot 4) - 4 = 0$

Step 1.6.1.2

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

$x^{2} + 8 x + 16 - (y^{2} + 4 \cdot y + 4 y + 4 \cdot 4) - 4 = 0$

Step 1.6.1.3

Multiply $4$ by $4$ .

$x^{2} + 8 x + 16 - (y^{2} + 4 y + 4 y + 16) - 4 = 0$

Step 1.6.2

Add $4 y$ and $4 y$ .

$x^{2} + 8 x + 16 - (y^{2} + 8 y + 16) - 4 = 0$

Step 1.7

Apply the distributive property.

$x^{2} + 8 x + 16 - y^{2} - (8 y) - 1 \cdot 16 - 4 = 0$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $8$ by $- 1$ .

$x^{2} + 8 x + 16 - y^{2} - 8 y - 1 \cdot 16 - 4 = 0$

Step 1.8.2

Multiply $- 1$ by $16$ .

$x^{2} + 8 x + 16 - y^{2} - 8 y - 16 - 4 = 0$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $x^{2} + 8 x + 16 - y^{2} - 8 y - 16 - 4$ .

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Step 2.1.1

Subtract $16$ from $16$ .

$x^{2} + 8 x - y^{2} - 8 y + 0 - 4 = 0$

Step 2.1.2

Add $x^{2} + 8 x - y^{2} - 8 y$ and $0$ .

$x^{2} + 8 x - y^{2} - 8 y - 4 = 0$

Step 2.2

Move $8 x$ .

$x^{2} - y^{2} + 8 x - 8 y - 4 = 0$

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