Find the Directrix (y plus square root of three) to the power of two equals minus four square root of two(x minus square root of two)

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

Reformulate the equation as the vertex.

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Isolate $x$ to the left side of the equation.

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Should be rewritten the equation as $- 4 \sqrt{2} (x - \sqrt{2}) = {(y + \sqrt{3})}^{2}$ .

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

First, divide each term by $- 4 \sqrt{2}$ and simplify.

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Divide each term in $- 4 \sqrt{2} (x - \sqrt{2}) = {(y + \sqrt{3})}^{2}$ by $- 4 \sqrt{2}$ .

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

Clarify the left side of the equation by cancelling the common factors.

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The common factor should be canceled of $- 4$ .

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Cancel the common factor.

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

Reformulate the expression.

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

The common factor $\sqrt{2}$ should be canceled.

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Cancel the common factor.

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

Divide $x - \sqrt{2}$ by $1$ .

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

Clarify $\frac{{(y + \sqrt{3})}^{2}}{- 4 \sqrt{2}}$ .

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First, multiply $\frac{{(y + \sqrt{3})}^{2}}{- 4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Add and Clarify the denominator.

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Multiply $\frac{{(y + \sqrt{3})}^{2}}{- 4 \sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Put $\sqrt{2}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Apply the power rule $a^{m} a^{n} = a^{m + n}$ to Add exponents.

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

Add $1$ and $1$ .

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

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

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Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Use the power rule when multiplying the exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Solve the exponent.

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

Clarify the expression.

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Multiply $- 4$ by $2$ .

$x - \sqrt{2} = \frac{{(y + \sqrt{3})}^{2} \sqrt{2}}{- 8}$

Put the negative in front of the fraction.

$x - \sqrt{2} = - \frac{{(y + \sqrt{3})}^{2} \sqrt{2}}{8}$

Add $\sqrt{2}$ to both sides of the equation.

$x = - \frac{{(y + \sqrt{3})}^{2} \sqrt{2}}{8} + \sqrt{2}$

Shift terms.

$x = - \frac{\sqrt{2}}{8} \cdot {(y + \sqrt{3})}^{2} + \sqrt{2}$

Complete the square for $- \frac{\sqrt{2}}{8} \cdot {(y + \sqrt{3})}^{2} + \sqrt{2}$ .

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Clarify each term.

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

$- \frac{\sqrt{2}}{8} \cdot ((y + \sqrt{3}) (y + \sqrt{3})) + \sqrt{2}$

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

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Use the distributive law.

$- \frac{\sqrt{2}}{8} \cdot (y (y + \sqrt{3}) + \sqrt{3} (y + \sqrt{3})) + \sqrt{2}$

Apply the distributive property.

$- \frac{\sqrt{2}}{8} \cdot (y \cdot y + y \sqrt{3} + \sqrt{3} (y + \sqrt{3})) + \sqrt{2}$

Apply the distributive property.

$- \frac{\sqrt{2}}{8} \cdot (y \cdot y + y \sqrt{3} + \sqrt{3} y + \sqrt{3} \sqrt{3}) + \sqrt{2}$

Clarify and combine like terms.

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Simplify each term.

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Multiply $y$ by $y$ .

$- \frac{\sqrt{2}}{8} \cdot (y^{2} + y \sqrt{3} + \sqrt{3} y + \sqrt{3} \sqrt{3}) + \sqrt{2}$

Add using the product rule for radicals.

$- \frac{\sqrt{2}}{8} \cdot (y^{2} + y \sqrt{3} + \sqrt{3} y + \sqrt{3 \cdot 3}) + \sqrt{2}$

Multiply $3$ by $3$ .

$- \frac{\sqrt{2}}{8} \cdot (y^{2} + y \sqrt{3} + \sqrt{3} y + \sqrt{9}) + \sqrt{2}$

Rewrite $9$ as $3^{2}$ .

$- \frac{\sqrt{2}}{8} \cdot (y^{2} + y \sqrt{3} + \sqrt{3} y + \sqrt{3^{2}}) + \sqrt{2}$

Withdraw terms from under the radical.

$- \frac{\sqrt{2}}{8} \cdot (y^{2} + y \sqrt{3} + \sqrt{3} y + | 3 |) + \sqrt{2}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$- \frac{\sqrt{2}}{8} \cdot (y^{2} + y \sqrt{3} + \sqrt{3} y + 3) + \sqrt{2}$

Shift the factors of $\sqrt{3} y$ .

$- \frac{\sqrt{2}}{8} \cdot (y^{2} + y \sqrt{3} + y \sqrt{3} + 3) + \sqrt{2}$

Add $y \sqrt{3}$ and $y \sqrt{3}$ .

$- \frac{\sqrt{2}}{8} \cdot (y^{2} + 2 y \sqrt{3} + 3) + \sqrt{2}$

Apply the distributive property.

$- \frac{\sqrt{2}}{8} y^{2} - \frac{\sqrt{2}}{8} (2 y \sqrt{3}) - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Simplify.

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Combine $y^{2}$ and $\frac{\sqrt{2}}{8}$ .

$- \frac{y^{2} \sqrt{2}}{8} - \frac{\sqrt{2}}{8} (2 y \sqrt{3}) - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{\sqrt{2}}{8}$ into the numerator.

$- \frac{y^{2} \sqrt{2}}{8} + \frac{- \sqrt{2}}{8} (2 y \sqrt{3}) - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Factor $2$ out of $8$ .

$- \frac{y^{2} \sqrt{2}}{8} + \frac{- \sqrt{2}}{2 (4)} (2 y \sqrt{3}) - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Factor $2$ out of $2 y \sqrt{3}$ .

$- \frac{y^{2} \sqrt{2}}{8} + \frac{- \sqrt{2}}{2 (4)} (2 (y \sqrt{3})) - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Cancel the common factor.

$- \frac{y^{2} \sqrt{2}}{8} + \frac{- \sqrt{2}}{2 \cdot 4} (2 (y \sqrt{3})) - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Rewrite the expression.

$- \frac{y^{2} \sqrt{2}}{8} + \frac{- \sqrt{2}}{4} (y \sqrt{3}) - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Combine $y$ and $\frac{- \sqrt{2}}{4}$ .

$- \frac{y^{2} \sqrt{2}}{8} + \frac{y (- \sqrt{2})}{4} \sqrt{3} - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Combine $\frac{y (- \sqrt{2})}{4}$ and $\sqrt{3}$ .

$- \frac{y^{2} \sqrt{2}}{8} + \frac{y (- \sqrt{2}) \sqrt{3}}{4} - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Combine using the product rule for radicals.

$- \frac{y^{2} \sqrt{2}}{8} + \frac{y (- \sqrt{3 \cdot 2})}{4} - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Multiply $3$ by $2$ .

$- \frac{y^{2} \sqrt{2}}{8} + \frac{y (- \sqrt{6})}{4} - \frac{\sqrt{2}}{8} \cdot 3 + \sqrt{2}$

Multiply $- \frac{\sqrt{2}}{8} \cdot 3$ .

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Multiply $3$ by $- 1$ .

$- \frac{y^{2} \sqrt{2}}{8} + \frac{y (- \sqrt{6})}{4} - 3 \frac{\sqrt{2}}{8} + \sqrt{2}$

Combine $- 3$ and $\frac{\sqrt{2}}{8}$ .

$- \frac{y^{2} \sqrt{2}}{8} + \frac{y (- \sqrt{6})}{4} + \frac{- 3 \sqrt{2}}{8} + \sqrt{2}$

Simplify each term.

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Factor out negative.

$- \frac{y^{2} \sqrt{2}}{8} + \frac{- y \sqrt{6}}{4} + \frac{- 3 \sqrt{2}}{8} + \sqrt{2}$

Move the negative in front of the fraction.

$- \frac{y^{2} \sqrt{2}}{8} - \frac{y \sqrt{6}}{4} + \frac{- 3 \sqrt{2}}{8} + \sqrt{2}$

Move the negative in front of the fraction.

$- \frac{y^{2} \sqrt{2}}{8} - \frac{y \sqrt{6}}{4} - \frac{3 \sqrt{2}}{8} + \sqrt{2}$

To write $\sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{y^{2} \sqrt{2}}{8} - \frac{y \sqrt{6}}{4} - \frac{3 \sqrt{2}}{8} + \sqrt{2} \cdot \frac{8}{8}$

Combine $\sqrt{2}$ and $\frac{8}{8}$ .

$- \frac{y^{2} \sqrt{2}}{8} - \frac{y \sqrt{6}}{4} - \frac{3 \sqrt{2}}{8} + \frac{\sqrt{2} \cdot 8}{8}$

Simplify the expression.

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Add numerators over the common denominator.

$- \frac{y^{2} \sqrt{2}}{8} - \frac{y \sqrt{6}}{4} + \frac{- 3 \sqrt{2} + \sqrt{2} \cdot 8}{8}$

Reorder the factors of $\sqrt{2} \cdot 8$ .

$- \frac{y^{2} \sqrt{2}}{8} - \frac{y \sqrt{6}}{4} + \frac{- 3 \sqrt{2} + 8 \sqrt{2}}{8}$

Add $- 3 \sqrt{2}$ and $8 \sqrt{2}$ .

$- \frac{y^{2} \sqrt{2}}{8} - \frac{y \sqrt{6}}{4} + \frac{5 \sqrt{2}}{8}$

Consider the $a x^{2} + b x + c$ form, to find the values of $a$ , $b$ , and $c$ .

$a = e r \cdot r o r, b = 0, c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Replace the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 (e r \cdot r o r)}$

Clarify the right side.

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Multiply $r$ by $r$ by adding the exponents.

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Move $r$ .

$d = \frac{0}{2 (e (r \cdot r) o r)}$

Multiply $r$ by $r$ .

$d = \frac{0}{2 (e r^{2} o r)}$

Multiply $r^{2}$ by $r$ by adding the exponents.

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Move $r$ .

$d = \frac{0}{2 (e (r \cdot r^{2}) o)}$

Multiply $r$ by $r^{2}$ .

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Raise $r$ to the power of $1$ .

$d = \frac{0}{2 (e (r \cdot r^{2}) o)}$

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

$d = \frac{0}{2 (e r^{1 + 2} o)}$

Add $1$ and $2$ .

$d = \frac{0}{2 (e r^{3} o)}$

The common factor $0$ should be canceled and $2$ .

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Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (e r^{3} o)}$

The common factors should be canceled.

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Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (e r^{3} o)}$

Rewrite the expression.

$d = \frac{0}{e r^{3} o}$

First, divide $0$ by $e r^{3} o$ .

$d = 0$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Subtract $\frac{{(0)}^{2}}{4 (e r \cdot r o r)}$ from $0$ .

$e = - \frac{{(0)}^{2}}{4 (e r \cdot r o r)}$

Multiply $r$ by $r$ by adding the exponents.

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Move $r$ .

$e = - \frac{{(0)}^{2}}{4 (e (r \cdot r) o r)}$

Multiply $r$ by $r$ .

$e = - \frac{{(0)}^{2}}{4 (e r^{2} o r)}$

Multiply $r^{2}$ by $r$ by adding the exponents.

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Move $r$ .

$e = - \frac{{(0)}^{2}}{4 (e (r \cdot r^{2}) o)}$

Multiply $r$ by $r^{2}$ .

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Raise $r$ to the power of $1$ .

$e = - \frac{{(0)}^{2}}{4 (e (r \cdot r^{2}) o)}$

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

$e = - \frac{{(0)}^{2}}{4 (e r^{1 + 2} o)}$

Add $1$ and $2$ .

$e = - \frac{{(0)}^{2}}{4 (e r^{3} o)}$

Raising $0$ to any positive power yields $0$ .

$e = - \frac{0}{4 e r^{3} o}$

Cancel the common factor of $0$ and $4$ .

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Factor $4$ out of $0$ .

$e = - \frac{4 (0)}{4 e r^{3} o}$

Cancel the common factors.

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Factor $4$ out of $4 e r^{3} o$ .

$e = - \frac{4 (0)}{4 (e r^{3} o)}$

Cancel the common factor.

$e = - \frac{4 \cdot 0}{4 (e r^{3} o)}$

Rewrite the expression.

$e = - \frac{0}{e r^{3} o}$

Multiply by zero.

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Divide $0$ by $e r^{3} o$ .

$e = - 0$

Multiply $- 1$ by $0$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $a {(x + d)}^{2} + e$ .

$- \frac{\sqrt{2}}{8} \cdot {(y + \sqrt{3})}^{2} + \sqrt{2}$

Set $x$ equal to the new right side.

$x = - \frac{\sqrt{2}}{8} \cdot {(y + \sqrt{3})}^{2} + \sqrt{2}$

Consider the vertex form, $x = a {(y - k)}^{2} + h$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{1}{8}$

$h = \sqrt{2}$

$k = - \sqrt{3}$

Find the vertex $(h, k)$ .

$(\sqrt{2}, - \sqrt{3})$

Calculate $p$ , the distance from the vertex to the focus.

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Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{8})}$

Simplify.

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The common factor $1$ should be canceled and $- 1$ .

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Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{8})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{8})}$

Combine $4$ and $\frac{1}{8}$ .

$- \frac{1}{\frac{4}{8}}$

The common factor $4$ should be canceled and $8$ .

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Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{8}}$

Cancel the common factors.

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Factor $4$ out of $8$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Rewrite the expression.

$- \frac{1}{\frac{1}{2}}$

Multiply the numerator by the denominator's reciprocal.

$- (1 \cdot 2)$

First, multiply $2$ by $1$ .

$- 1 \cdot 2$

Multiply $- 1$ by $2$ .

$- 2$

Find the directrix.

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The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Substitute the known values of $p$ and $h$ into the formula and simplify.

$x = \sqrt{2} + 2$

The result can be displayed in a variety of ways.

Exact Form:

$x = \sqrt{2} + 2$

Decimal Form:

$x = 3.41421356 \dots$

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