Examples

{(x + 2)}^{2} - {(y - \sqrt{5})}^{2} + 4 = 0

${(x + 2)}^{2} - {(y - \sqrt{5})}^{2} + 4 = 0$

Step 1

Simplify each term.

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Step 1.1

Rewrite

{(x + 2)}^{2}

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

(x + 2) (x + 2)

$(x + 2) (x + 2)$ .

(x + 2) (x + 2) - {(y - \sqrt{5})}^{2} + 4 = 0

$(x + 2) (x + 2) - {(y - \sqrt{5})}^{2} + 4 = 0$

Step 1.2

Expand

(x + 2) (x + 2)

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

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Step 1.2.1

Apply the distributive property.

x (x + 2) + 2 (x + 2) - {(y - \sqrt{5})}^{2} + 4 = 0

$x (x + 2) + 2 (x + 2) - {(y - \sqrt{5})}^{2} + 4 = 0$

Step 1.2.2

Apply the distributive property.

x \cdot x + x \cdot 2 + 2 (x + 2) - {(y - \sqrt{5})}^{2} + 4 = 0

$x \cdot x + x \cdot 2 + 2 (x + 2) - {(y - \sqrt{5})}^{2} + 4 = 0$

Step 1.2.3

Apply the distributive property.

x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 - {(y - \sqrt{5})}^{2} + 4 = 0

$x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 - {(y - \sqrt{5})}^{2} + 4 = 0$

x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 - {(y - \sqrt{5})}^{2} + 4 = 0

$x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 - {(y - \sqrt{5})}^{2} + 4 = 0$

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

$x$ by

x

$x$ .

x^{2} + x \cdot 2 + 2 x + 2 \cdot 2 - {(y - \sqrt{5})}^{2} + 4 = 0

$x^{2} + x \cdot 2 + 2 x + 2 \cdot 2 - {(y - \sqrt{5})}^{2} + 4 = 0$

Step 1.3.1.2

Move

2

$2$ to the left of

x

$x$ .

x^{2} + 2 \cdot x + 2 x + 2 \cdot 2 - {(y - \sqrt{5})}^{2} + 4 = 0

$x^{2} + 2 \cdot x + 2 x + 2 \cdot 2 - {(y - \sqrt{5})}^{2} + 4 = 0$

Step 1.3.1.3

Multiply

2

$2$ by

2

$2$ .

x^{2} + 2 x + 2 x + 4 - {(y - \sqrt{5})}^{2} + 4 = 0

$x^{2} + 2 x + 2 x + 4 - {(y - \sqrt{5})}^{2} + 4 = 0$

x^{2} + 2 x + 2 x + 4 - {(y - \sqrt{5})}^{2} + 4 = 0

$x^{2} + 2 x + 2 x + 4 - {(y - \sqrt{5})}^{2} + 4 = 0$

Step 1.3.2

Add

2 x

$2 x$ and

2 x

$2 x$ .

x^{2} + 4 x + 4 - {(y - \sqrt{5})}^{2} + 4 = 0

$x^{2} + 4 x + 4 - {(y - \sqrt{5})}^{2} + 4 = 0$

x^{2} + 4 x + 4 - {(y - \sqrt{5})}^{2} + 4 = 0

$x^{2} + 4 x + 4 - {(y - \sqrt{5})}^{2} + 4 = 0$

Step 1.4

Rewrite

{(y - \sqrt{5})}^{2}

${(y - \sqrt{5})}^{2}$ as

(y - \sqrt{5}) (y - \sqrt{5})

$(y - \sqrt{5}) (y - \sqrt{5})$ .

x^{2} + 4 x + 4 - ((y - \sqrt{5}) (y - \sqrt{5})) + 4 = 0

$x^{2} + 4 x + 4 - ((y - \sqrt{5}) (y - \sqrt{5})) + 4 = 0$

Step 1.5

Expand

(y - \sqrt{5}) (y - \sqrt{5})

$(y - \sqrt{5}) (y - \sqrt{5})$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

x^{2} + 4 x + 4 - (y (y - \sqrt{5}) - \sqrt{5} (y - \sqrt{5})) + 4 = 0

$x^{2} + 4 x + 4 - (y (y - \sqrt{5}) - \sqrt{5} (y - \sqrt{5})) + 4 = 0$

Step 1.5.2

Apply the distributive property.

x^{2} + 4 x + 4 - (y \cdot y + y (- \sqrt{5}) - \sqrt{5} (y - \sqrt{5})) + 4 = 0

$x^{2} + 4 x + 4 - (y \cdot y + y (- \sqrt{5}) - \sqrt{5} (y - \sqrt{5})) + 4 = 0$

Step 1.5.3

Apply the distributive property.

x^{2} + 4 x + 4 - (y \cdot y + y (- \sqrt{5}) - \sqrt{5} y - \sqrt{5} (- \sqrt{5})) + 4 = 0

$x^{2} + 4 x + 4 - (y \cdot y + y (- \sqrt{5}) - \sqrt{5} y - \sqrt{5} (- \sqrt{5})) + 4 = 0$

x^{2} + 4 x + 4 - (y \cdot y + y (- \sqrt{5}) - \sqrt{5} y - \sqrt{5} (- \sqrt{5})) + 4 = 0

$x^{2} + 4 x + 4 - (y \cdot y + y (- \sqrt{5}) - \sqrt{5} y - \sqrt{5} (- \sqrt{5})) + 4 = 0$

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

$y$ by

y

$y$ .

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y - \sqrt{5} (- \sqrt{5})) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y - \sqrt{5} (- \sqrt{5})) + 4 = 0$

Step 1.6.1.2

Multiply

- \sqrt{5} (- \sqrt{5})

$- \sqrt{5} (- \sqrt{5})$ .

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Step 1.6.1.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 1 \sqrt{5} \sqrt{5}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 1 \sqrt{5} \sqrt{5}) + 4 = 0$

Step 1.6.1.2.2

Multiply

\sqrt{5}

$\sqrt{5}$ by

1

$1$ .

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + \sqrt{5} \sqrt{5}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + \sqrt{5} \sqrt{5}) + 4 = 0$

Step 1.6.1.2.3

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + \sqrt{5} \sqrt{5}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + \sqrt{5} \sqrt{5}) + 4 = 0$

Step 1.6.1.2.4

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + \sqrt{5} \sqrt{5}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + \sqrt{5} \sqrt{5}) + 4 = 0$

Step 1.6.1.2.5

Use the power rule

a^{m} a^{n} = a^{m + n}

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

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + {\sqrt{5}}^{1 + 1}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + {\sqrt{5}}^{1 + 1}) + 4 = 0$

Step 1.6.1.2.6

Add

1

$1$ and

1

$1$ .

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + {\sqrt{5}}^{2}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + {\sqrt{5}}^{2}) + 4 = 0$

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + {\sqrt{5}}^{2}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + {\sqrt{5}}^{2}) + 4 = 0$

Step 1.6.1.3

Rewrite

{\sqrt{5}}^{2}

${\sqrt{5}}^{2}$ as

5

$5$ .

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Step 1.6.1.3.1

Use

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

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

\sqrt{5}

$\sqrt{5}$ as

5^{\frac{1}{2}}

$5^{\frac{1}{2}}$ .

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + {(5^{\frac{1}{2}})}^{2}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + {(5^{\frac{1}{2}})}^{2}) + 4 = 0$

Step 1.6.1.3.2

Apply the power rule and multiply exponents,

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

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

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 5^{\frac{1}{2} \cdot 2}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 5^{\frac{1}{2} \cdot 2}) + 4 = 0$

Step 1.6.1.3.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 5^{\frac{2}{2}}) + 4 = 0

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 5^{\frac{2}{2}}) + 4 = 0$

Step 1.6.1.3.4

Cancel the common factor of

2

$2$ .

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Step 1.6.1.3.4.1

Cancel the common factor.

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 5^{\frac{2}{2}}) + 4 = 0$

Step 1.6.1.3.4.2

Rewrite the expression.

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 5) + 4 = 0$

Step 1.6.1.3.5

Evaluate the exponent.

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - \sqrt{5} y + 5) + 4 = 0$

Step 1.6.2

Reorder the factors of

$- \sqrt{5} y$ .

$x^{2} + 4 x + 4 - (y^{2} + y (- \sqrt{5}) - y \sqrt{5} + 5) + 4 = 0$

Step 1.6.3

Subtract

$y \sqrt{5}$ from

$y (- \sqrt{5})$ .

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Step 1.6.3.1

Reorder

$y$ and

$- 1$ .

$x^{2} + 4 x + 4 - (y^{2} - 1 \cdot (y \sqrt{5}) - y \sqrt{5} + 5) + 4 = 0$

Step 1.6.3.2

Subtract

$y \sqrt{5}$ from

$- 1 \cdot y \sqrt{5}$ .

$x^{2} + 4 x + 4 - (y^{2} - 2 y \sqrt{5} + 5) + 4 = 0$

Step 1.7

Apply the distributive property.

$x^{2} + 4 x + 4 - y^{2} - (- 2 y \sqrt{5}) - 1 \cdot 5 + 4 = 0$

Step 1.8

Simplify.

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Step 1.8.1

Multiply

$- 2$ by

$- 1$ .

$x^{2} + 4 x + 4 - y^{2} + 2 (y \sqrt{5}) - 1 \cdot 5 + 4 = 0$

Step 1.8.2

Multiply

$- 1$ by

$5$ .

$x^{2} + 4 x + 4 - y^{2} + 2 (y \sqrt{5}) - 5 + 4 = 0$

$x^{2} + 4 x + 4 - y^{2} + 2 y \sqrt{5} - 5 + 4 = 0$

Step 2

Simplify the expression.

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Step 2.1

Subtract

$5$ from

$4$ .

$x^{2} + 4 x - y^{2} + 2 y \sqrt{5} - 1 + 4 = 0$

Step 2.2

Add

$- 1$ and

$4$ .

$x^{2} + 4 x - y^{2} + 2 y \sqrt{5} + 3 = 0$

Step 2.3

Move

$4 x$ .

$x^{2} - y^{2} + 4 x + 2 y \sqrt{5} + 3 = 0$

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