Precalculus Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

Multiply $- 6$ by $- 6$ .

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

Step 1.1.3.2

Subtract $6 x$ from $- 6 x$ .

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

Step 1.1.4

Apply the distributive property.

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

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply $- 12$ by $2$ .

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

Step 1.1.5.2

Multiply $2$ by $36$ .

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

Step 1.1.6

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

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

Step 1.1.7

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

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Step 1.1.7.1

Apply the distributive property.

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

Step 1.1.7.2

Apply the distributive property.

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

Step 1.1.7.3

Apply the distributive property.

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

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 $y$ by $y$ .

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

Step 1.1.8.1.2

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

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

Step 1.1.8.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.1.8.2

Subtract $2 y$ from $- 2 y$ .

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

Step 1.1.9

Apply the distributive property.

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

Step 1.1.10

Simplify.

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Step 1.1.10.1

Multiply $- 4$ by $- 1$ .

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

Step 1.1.10.2

Multiply $- 1$ by $4$ .

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Subtract $4$ from $72$ .

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

Step 1.2.2

Move $- 24 x$ .

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

Step 2

Set the equation equal to zero.

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Step 2.1

Subtract $16$ from both sides of the equation.

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

Step 2.2

Subtract $16$ from $68$ .

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

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