Examples

${(x - \sqrt{4})}^{2} - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

$4$ as

$2^{2}$ .

${(x - \sqrt{2^{2}})}^{2} - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.1.2

Pull terms out from under the radical, assuming positive real numbers.

${(x - 1 \cdot 2)}^{2} - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.1.3

Multiply

$- 1$ by

$2$ .

${(x - 2)}^{2} - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.2

Rewrite

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

$(x - 2) (x - 2)$ .

$(x - 2) (x - 2) - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.3

Expand

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

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Step 1.3.1

Apply the distributive property.

$x (x - 2) - 2 (x - 2) - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.3.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 (x - 2) - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.3.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply

$x$ by

$x$ .

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.4.1.2

Move

$- 2$ to the left of

$x$ .

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.4.1.3

Multiply

$- 2$ by

$- 2$ .

$x^{2} - 2 x - 2 x + 4 - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.4.2

Subtract

$2 x$ from

$- 2 x$ .

$x^{2} - 4 x + 4 - {(y + 3 \sqrt{2})}^{2} - 4 = 0$

Step 1.5

Rewrite

${(y + 3 \sqrt{2})}^{2}$ as

$(y + 3 \sqrt{2}) (y + 3 \sqrt{2})$ .

$x^{2} - 4 x + 4 - ((y + 3 \sqrt{2}) (y + 3 \sqrt{2})) - 4 = 0$

Step 1.6

Expand

$(y + 3 \sqrt{2}) (y + 3 \sqrt{2})$ using the FOIL Method.

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Step 1.6.1

Apply the distributive property.

$x^{2} - 4 x + 4 - (y (y + 3 \sqrt{2}) + 3 \sqrt{2} (y + 3 \sqrt{2})) - 4 = 0$

Step 1.6.2

Apply the distributive property.

$x^{2} - 4 x + 4 - (y \cdot y + y (3 \sqrt{2}) + 3 \sqrt{2} (y + 3 \sqrt{2})) - 4 = 0$

Step 1.6.3

Apply the distributive property.

$x^{2} - 4 x + 4 - (y \cdot y + y (3 \sqrt{2}) + 3 \sqrt{2} y + 3 \sqrt{2} (3 \sqrt{2})) - 4 = 0$

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply

$y$ by

$y$ .

$x^{2} - 4 x + 4 - (y^{2} + y (3 \sqrt{2}) + 3 \sqrt{2} y + 3 \sqrt{2} (3 \sqrt{2})) - 4 = 0$

Step 1.7.1.2

Move

$3$ to the left of

$y$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 \cdot (y \sqrt{2}) + 3 \sqrt{2} y + 3 \sqrt{2} (3 \sqrt{2})) - 4 = 0$

Step 1.7.1.3

Multiply

$3 \sqrt{2} (3 \sqrt{2})$ .

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Step 1.7.1.3.1

Multiply

$3$ by

$3$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 \sqrt{2} \sqrt{2}) - 4 = 0$

Step 1.7.1.3.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 (\sqrt{2} \sqrt{2})) - 4 = 0$

Step 1.7.1.3.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 (\sqrt{2} \sqrt{2})) - 4 = 0$

Step 1.7.1.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 {\sqrt{2}}^{1 + 1}) - 4 = 0$

Step 1.7.1.3.5

Add

$1$ and

$1$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 {\sqrt{2}}^{2}) - 4 = 0$

Step 1.7.1.4

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 1.7.1.4.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 {(2^{\frac{1}{2}})}^{2}) - 4 = 0$

Step 1.7.1.4.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 \cdot 2^{\frac{1}{2} \cdot 2}) - 4 = 0$

Step 1.7.1.4.3

Combine

$\frac{1}{2}$ and

$2$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 \cdot 2^{\frac{2}{2}}) - 4 = 0$

Step 1.7.1.4.4

Cancel the common factor of

$2$ .

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Step 1.7.1.4.4.1

Cancel the common factor.

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 \cdot 2^{\frac{2}{2}}) - 4 = 0$

Step 1.7.1.4.4.2

Rewrite the expression.

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 \cdot 2) - 4 = 0$

Step 1.7.1.4.5

Evaluate the exponent.

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 9 \cdot 2) - 4 = 0$

Step 1.7.1.5

Multiply

$9$ by

$2$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 \sqrt{2} y + 18) - 4 = 0$

Step 1.7.2

Reorder the factors of

$3 \sqrt{2} y$ .

$x^{2} - 4 x + 4 - (y^{2} + 3 y \sqrt{2} + 3 y \sqrt{2} + 18) - 4 = 0$

Step 1.7.3

Add

$3 y \sqrt{2}$ and

$3 y \sqrt{2}$ .

$x^{2} - 4 x + 4 - (y^{2} + 6 y \sqrt{2} + 18) - 4 = 0$

Step 1.8

Apply the distributive property.

$x^{2} - 4 x + 4 - y^{2} - (6 y \sqrt{2}) - 1 \cdot 18 - 4 = 0$

Step 1.9

Simplify.

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Step 1.9.1

Multiply

$6$ by

$- 1$ .

$x^{2} - 4 x + 4 - y^{2} - 6 (y \sqrt{2}) - 1 \cdot 18 - 4 = 0$

Step 1.9.2

Multiply

$- 1$ by

$18$ .

$x^{2} - 4 x + 4 - y^{2} - 6 (y \sqrt{2}) - 18 - 4 = 0$

$x^{2} - 4 x + 4 - y^{2} - 6 y \sqrt{2} - 18 - 4 = 0$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in

$x^{2} - 4 x + 4 - y^{2} - 6 y \sqrt{2} - 18 - 4$ .

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Step 2.1.1

Subtract

$4$ from

$4$ .

$x^{2} - 4 x - y^{2} - 6 y \sqrt{2} - 18 + 0 = 0$

Step 2.1.2

Add

$x^{2} - 4 x - y^{2} - 6 y \sqrt{2} - 18$ and

$0$ .

$x^{2} - 4 x - y^{2} - 6 y \sqrt{2} - 18 = 0$

Step 2.2

Move

$- 4 x$ .

$x^{2} - y^{2} - 4 x - 6 y \sqrt{2} - 18 = 0$

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