Algebra Examples

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

Simplify each term.

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

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

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

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

Multiply $2$ by $2$ .

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

Add $2 x$ and $2 x$ .

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

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $y$ by $y$ .

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

Move $- 1$ to the left of $y$ .

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

Rewrite $- 1 y$ as $- y$ .

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

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

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Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Evaluate the exponent.

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

Reorder the factors of $- \sqrt{5} y$ .

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

Subtract $y \sqrt{5}$ from $- y \sqrt{5}$ .

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

Simplify by adding terms.

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

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Subtract $4$ from $4$ .

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

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

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

Move $4 x$ .

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

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