Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 1.3.1.2

Move $2$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.2

Add $2 x$ and $2 x$ .

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

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 $y$ .

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

Step 1.6.1.2

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

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Step 1.6.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.2.2

Multiply $\sqrt{5}$ by $1$ .

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

Step 1.6.1.2.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 1.6.1.2.4

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 1.6.1.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.6.1.2.6

Add $1$ and $1$ .

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

Step 1.6.1.3

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

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Step 1.6.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 1.6.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.6.1.3.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.6.1.3.4

Cancel the common factor of $2$ .

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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 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $x^{2} + 4 x + 4 + y^{2} - 2 y \sqrt{5} + 5 - 4$ .

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Step 2.1.1

Subtract $4$ from $4$ .

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

Step 2.1.2

Add $x^{2} + 4 x + y^{2} - 2 y \sqrt{5} + 5$ and $0$ .

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

Step 2.2

Move $4 x$ .

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

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