Examples

2 {(x - 6)}^{2} - {(y - 2)}^{2} = 16

$2 {(x - 6)}^{2} - {(y - 2)}^{2} = 16$

Step 1

Simplify the left side

2 {(x - 6)}^{2} - {(y - 2)}^{2}

$2 {(x - 6)}^{2} - {(y - 2)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

{(x - 6)}^{2}

${(x - 6)}^{2}$ as

(x - 6) (x - 6)

$(x - 6) (x - 6)$ .

2 ((x - 6) (x - 6)) - {(y - 2)}^{2} = 16

$2 ((x - 6) (x - 6)) - {(y - 2)}^{2} = 16$

Step 1.1.2

Expand

(x - 6) (x - 6)

$(x - 6) (x - 6)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

2 (x (x - 6) - 6 (x - 6)) - {(y - 2)}^{2} = 16

$2 (x (x - 6) - 6 (x - 6)) - {(y - 2)}^{2} = 16$

Step 1.1.2.2

Apply the distributive property.

2 (x \cdot x + x \cdot - 6 - 6 (x - 6)) - {(y - 2)}^{2} = 16

$2 (x \cdot x + x \cdot - 6 - 6 (x - 6)) - {(y - 2)}^{2} = 16$

Step 1.1.2.3

Apply the distributive property.

2 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) - {(y - 2)}^{2} = 16

$2 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) - {(y - 2)}^{2} = 16$

2 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) - {(y - 2)}^{2} = 16

$2 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) - {(y - 2)}^{2} = 16$

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

$x$ by

x

$x$ .

2 (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) - {(y - 2)}^{2} = 16

$2 (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) - {(y - 2)}^{2} = 16$

Step 1.1.3.1.2

Move

- 6

$- 6$ to the left of

x

$x$ .

2 (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) - {(y - 2)}^{2} = 16

$2 (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) - {(y - 2)}^{2} = 16$

Step 1.1.3.1.3

Multiply

- 6

$- 6$ by

- 6

$- 6$ .

2 (x^{2} - 6 x - 6 x + 36) - {(y - 2)}^{2} = 16

$2 (x^{2} - 6 x - 6 x + 36) - {(y - 2)}^{2} = 16$

2 (x^{2} - 6 x - 6 x + 36) - {(y - 2)}^{2} = 16

$2 (x^{2} - 6 x - 6 x + 36) - {(y - 2)}^{2} = 16$

Step 1.1.3.2

Subtract

6 x

$6 x$ from

- 6 x

$- 6 x$ .

2 (x^{2} - 12 x + 36) - {(y - 2)}^{2} = 16

$2 (x^{2} - 12 x + 36) - {(y - 2)}^{2} = 16$

2 (x^{2} - 12 x + 36) - {(y - 2)}^{2} = 16

$2 (x^{2} - 12 x + 36) - {(y - 2)}^{2} = 16$

Step 1.1.4

Apply the distributive property.

2 x^{2} + 2 (- 12 x) + 2 \cdot 36 - {(y - 2)}^{2} = 16

$2 x^{2} + 2 (- 12 x) + 2 \cdot 36 - {(y - 2)}^{2} = 16$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply

- 12

$- 12$ by

2

$2$ .

2 x^{2} - 24 x + 2 \cdot 36 - {(y - 2)}^{2} = 16

$2 x^{2} - 24 x + 2 \cdot 36 - {(y - 2)}^{2} = 16$

Step 1.1.5.2

Multiply

2

$2$ by

36

$36$ .

2 x^{2} - 24 x + 72 - {(y - 2)}^{2} = 16

$2 x^{2} - 24 x + 72 - {(y - 2)}^{2} = 16$

2 x^{2} - 24 x + 72 - {(y - 2)}^{2} = 16

$2 x^{2} - 24 x + 72 - {(y - 2)}^{2} = 16$

Step 1.1.6

Rewrite

{(y - 2)}^{2}

${(y - 2)}^{2}$ as

(y - 2) (y - 2)

$(y - 2) (y - 2)$ .

2 x^{2} - 24 x + 72 - ((y - 2) (y - 2)) = 16

$2 x^{2} - 24 x + 72 - ((y - 2) (y - 2)) = 16$

Step 1.1.7

Expand

(y - 2) (y - 2)

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

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Step 1.1.7.1

Apply the distributive property.

2 x^{2} - 24 x + 72 - (y (y - 2) - 2 (y - 2)) = 16

$2 x^{2} - 24 x + 72 - (y (y - 2) - 2 (y - 2)) = 16$

Step 1.1.7.2

Apply the distributive property.

2 x^{2} - 24 x + 72 - (y \cdot y + y \cdot - 2 - 2 (y - 2)) = 16

$2 x^{2} - 24 x + 72 - (y \cdot y + y \cdot - 2 - 2 (y - 2)) = 16$

Step 1.1.7.3

Apply the distributive property.

2 x^{2} - 24 x + 72 - (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) = 16

$2 x^{2} - 24 x + 72 - (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) = 16$

2 x^{2} - 24 x + 72 - (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) = 16

$2 x^{2} - 24 x + 72 - (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) = 16$

Step 1.1.8

Simplify and combine like terms.

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

Multiply

y

$y$ by

y

$y$ .

2 x^{2} - 24 x + 72 - (y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2) = 16

$2 x^{2} - 24 x + 72 - (y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2) = 16$

Step 1.1.8.1.2

Move

- 2

$- 2$ to the left of

y

$y$ .

2 x^{2} - 24 x + 72 - (y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2) = 16

$2 x^{2} - 24 x + 72 - (y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2) = 16$

Step 1.1.8.1.3

Multiply

- 2

$- 2$ by

- 2

$- 2$ .

2 x^{2} - 24 x + 72 - (y^{2} - 2 y - 2 y + 4) = 16

$2 x^{2} - 24 x + 72 - (y^{2} - 2 y - 2 y + 4) = 16$

2 x^{2} - 24 x + 72 - (y^{2} - 2 y - 2 y + 4) = 16

$2 x^{2} - 24 x + 72 - (y^{2} - 2 y - 2 y + 4) = 16$

Step 1.1.8.2

Subtract

2 y

$2 y$ from

- 2 y

$- 2 y$ .

2 x^{2} - 24 x + 72 - (y^{2} - 4 y + 4) = 16

$2 x^{2} - 24 x + 72 - (y^{2} - 4 y + 4) = 16$

2 x^{2} - 24 x + 72 - (y^{2} - 4 y + 4) = 16

$2 x^{2} - 24 x + 72 - (y^{2} - 4 y + 4) = 16$

Step 1.1.9

Apply the distributive property.

2 x^{2} - 24 x + 72 - y^{2} - (- 4 y) - 1 \cdot 4 = 16

$2 x^{2} - 24 x + 72 - y^{2} - (- 4 y) - 1 \cdot 4 = 16$

Step 1.1.10

Simplify.

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Step 1.1.10.1

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

2 x^{2} - 24 x + 72 - y^{2} + 4 y - 1 \cdot 4 = 16

$2 x^{2} - 24 x + 72 - y^{2} + 4 y - 1 \cdot 4 = 16$

Step 1.1.10.2

Multiply

- 1

$- 1$ by

4

$4$ .

2 x^{2} - 24 x + 72 - y^{2} + 4 y - 4 = 16

$2 x^{2} - 24 x + 72 - y^{2} + 4 y - 4 = 16$

2 x^{2} - 24 x + 72 - y^{2} + 4 y - 4 = 16

$2 x^{2} - 24 x + 72 - y^{2} + 4 y - 4 = 16$

2 x^{2} - 24 x + 72 - y^{2} + 4 y - 4 = 16

$2 x^{2} - 24 x + 72 - y^{2} + 4 y - 4 = 16$

Step 1.2

Simplify the expression.

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Step 1.2.1

Subtract

4

$4$ from

72

$72$ .

2 x^{2} - 24 x - y^{2} + 4 y + 68 = 16

$2 x^{2} - 24 x - y^{2} + 4 y + 68 = 16$

Step 1.2.2

Move

- 24 x

$- 24 x$ .

2 x^{2} - y^{2} - 24 x + 4 y + 68 = 16

$2 x^{2} - y^{2} - 24 x + 4 y + 68 = 16$

2 x^{2} - y^{2} - 24 x + 4 y + 68 = 16

$2 x^{2} - y^{2} - 24 x + 4 y + 68 = 16$

2 x^{2} - y^{2} - 24 x + 4 y + 68 = 16

$2 x^{2} - y^{2} - 24 x + 4 y + 68 = 16$

Step 2

Set the equation equal to zero.

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Step 2.1

Subtract

16

$16$ from both sides of the equation.

2 x^{2} - y^{2} - 24 x + 4 y + 68 - 16 = 0

$2 x^{2} - y^{2} - 24 x + 4 y + 68 - 16 = 0$

Step 2.2

Subtract

16

$16$ from

68

$68$ .

2 x^{2} - y^{2} - 24 x + 4 y + 52 = 0

$2 x^{2} - y^{2} - 24 x + 4 y + 52 = 0$

2 x^{2} - y^{2} - 24 x + 4 y + 52 = 0

$2 x^{2} - y^{2} - 24 x + 4 y + 52 = 0$

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