Precalculus Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $4$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Rewrite the expression.

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

Step 1.1.2.2.2

Cancel the common factor of $\sqrt{2}$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2.2

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

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

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

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

Step 1.1.2.3.2

Combine and simplify the denominator.

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Step 1.1.2.3.2.1

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

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

Step 1.1.2.3.2.2

Move $\sqrt{2}$ .

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

Step 1.1.2.3.2.3

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

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

Step 1.1.2.3.2.4

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

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

Step 1.1.2.3.2.5

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

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

Step 1.1.2.3.2.6

Add $1$ and $1$ .

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

Step 1.1.2.3.2.7

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

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Step 1.1.2.3.2.7.1

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

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

Step 1.1.2.3.2.7.2

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

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

Step 1.1.2.3.2.7.3

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

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

Step 1.1.2.3.2.7.4

Cancel the common factor of $2$ .

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Step 1.1.2.3.2.7.4.1

Cancel the common factor.

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

Step 1.1.2.3.2.7.4.2

Rewrite the expression.

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

Step 1.1.2.3.2.7.5

Evaluate the exponent.

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

Step 1.1.2.3.3

Multiply $4$ by $2$ .

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

Step 1.1.3

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

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

Step 1.1.4

Reorder terms.

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

Step 1.2

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

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

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

Step 1.2.1.1.2

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

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Step 1.2.1.1.2.1

Apply the distributive property.

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

Step 1.2.1.1.2.2

Apply the distributive property.

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

Step 1.2.1.1.2.3

Apply the distributive property.

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

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $y$ by $y$ .

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

Step 1.2.1.1.3.1.2

Combine using the product rule for radicals.

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

Step 1.2.1.1.3.1.3

Multiply $3$ by $3$ .

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

Step 1.2.1.1.3.1.4

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

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

Step 1.2.1.1.3.1.5

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

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

Step 1.2.1.1.3.2

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

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

Step 1.2.1.1.3.3

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

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

Step 1.2.1.1.4

Apply the distributive property.

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

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

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

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

Step 1.2.1.1.5.2

Cancel the common factor of $2$ .

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Step 1.2.1.1.5.2.1

Factor $2$ out of $8$ .

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

Step 1.2.1.1.5.2.2

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

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

Step 1.2.1.1.5.2.3

Cancel the common factor.

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

Step 1.2.1.1.5.2.4

Rewrite the expression.

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

Step 1.2.1.1.5.3

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

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

Step 1.2.1.1.5.4

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

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

Step 1.2.1.1.5.5

Combine using the product rule for radicals.

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

Step 1.2.1.1.5.6

Multiply $3$ by $2$ .

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

Step 1.2.1.1.5.7

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

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

Step 1.2.1.1.6

Move $3$ to the left of $\sqrt{2}$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Combine the numerators over the common denominator.

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

Step 1.2.1.5

Combine the numerators over the common denominator.

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

Step 1.2.1.6

Move $8$ to the left of $\sqrt{2}$ .

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

Step 1.2.1.7

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

$\frac{\sqrt{2} y^{2} + 11 \sqrt{2}}{8} + \frac{y \sqrt{6}}{4}$

Step 1.2.1.8

To write $\frac{y \sqrt{6}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{2} y^{2} + 11 \sqrt{2}}{8} + \frac{y \sqrt{6}}{4} \cdot \frac{2}{2}$

Step 1.2.1.9

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1.9.1

Multiply $\frac{y \sqrt{6}}{4}$ by $\frac{2}{2}$ .

$\frac{\sqrt{2} y^{2} + 11 \sqrt{2}}{8} + \frac{y \sqrt{6} \cdot 2}{4 \cdot 2}$

Step 1.2.1.9.2

Multiply $4$ by $2$ .

$\frac{\sqrt{2} y^{2} + 11 \sqrt{2}}{8} + \frac{y \sqrt{6} \cdot 2}{8}$

Step 1.2.1.10

Combine the numerators over the common denominator.

$\frac{\sqrt{2} y^{2} + 11 \sqrt{2} + y \sqrt{6} \cdot 2}{8}$

Step 1.2.1.11

Simplify the numerator.

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Step 1.2.1.11.1

Move $2$ to the left of $y \sqrt{6}$ .

$\frac{\sqrt{2} y^{2} + 11 \sqrt{2} + 2 \cdot (y \sqrt{6})}{8}$

Step 1.2.1.11.2

Reorder terms.

$\frac{y^{2} \sqrt{2} + 2 y \sqrt{6} + 11 \sqrt{2}}{8}$

Step 1.2.2

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

$a = \frac{\sqrt{2}}{8}$

$b = \frac{\sqrt{6}}{4}$

$c = \frac{11 \sqrt{2}}{8}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.4.1

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

$d = \frac{\frac{\sqrt{6}}{4}}{2 \frac{\sqrt{2}}{8}}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{\sqrt{6}}{4} \cdot \frac{1}{2 \frac{\sqrt{2}}{8}}$

Step 1.2.4.2.2

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

$d = \frac{\sqrt{6}}{4} \cdot \frac{1}{\frac{2 \sqrt{2}}{8}}$

Step 1.2.4.2.3

Reduce the expression $\frac{2 \sqrt{2}}{8}$ by cancelling the common factors.

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Step 1.2.4.2.3.1

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

$d = \frac{\sqrt{6}}{4} \cdot \frac{1}{\frac{2 (\sqrt{2})}{8}}$

Step 1.2.4.2.3.2

Factor $2$ out of $8$ .

$d = \frac{\sqrt{6}}{4} \cdot \frac{1}{\frac{2 \sqrt{2}}{2 \cdot 4}}$

Step 1.2.4.2.3.3

Cancel the common factor.

$d = \frac{\sqrt{6}}{4} \cdot \frac{1}{\frac{2 \sqrt{2}}{2 \cdot 4}}$

Step 1.2.4.2.3.4

Rewrite the expression.

$d = \frac{\sqrt{6}}{4} \cdot \frac{1}{\frac{\sqrt{2}}{4}}$

Step 1.2.4.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{\sqrt{6}}{4} (1 \frac{4}{\sqrt{2}})$

Step 1.2.4.2.5

Multiply $\frac{4}{\sqrt{2}}$ by $1$ .

$d = \frac{\sqrt{6}}{4} \cdot \frac{4}{\sqrt{2}}$

Step 1.2.4.2.6

Cancel the common factor of $4$ .

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Step 1.2.4.2.6.1

Cancel the common factor.

$d = \frac{\sqrt{6}}{4} \cdot \frac{4}{\sqrt{2}}$

Step 1.2.4.2.6.2

Rewrite the expression.

$d = \sqrt{6} \frac{1}{\sqrt{2}}$

Step 1.2.4.2.7

Combine $\sqrt{6}$ and $\frac{1}{\sqrt{2}}$ .

$d = \frac{\sqrt{6}}{\sqrt{2}}$

Step 1.2.4.2.8

Combine $\sqrt{6}$ and $\sqrt{2}$ into a single radical.

$d = \sqrt{\frac{6}{2}}$

Step 1.2.4.2.9

Divide $6$ by $2$ .

$d = \sqrt{3}$

Step 1.2.5

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

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Step 1.2.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{{(\frac{\sqrt{6}}{4})}^{2}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

Apply the product rule to $\frac{\sqrt{6}}{4}$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{{\sqrt{6}}^{2}}{4^{2}}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 1.2.5.2.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{{(6^{\frac{1}{2}})}^{2}}{4^{2}}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.2.2

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

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{6^{\frac{1}{2} \cdot 2}}{4^{2}}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.2.3

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

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{6^{\frac{2}{2}}}{4^{2}}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.2.4

Cancel the common factor of $2$ .

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Step 1.2.5.2.1.1.2.4.1

Cancel the common factor.

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{6^{\frac{2}{2}}}{4^{2}}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.2.4.2

Rewrite the expression.

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{6^{1}}{4^{2}}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.2.5

Evaluate the exponent.

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{6}{4^{2}}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.3

Raise $4$ to the power of $2$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{6}{16}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.4

Cancel the common factor of $6$ and $16$ .

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Step 1.2.5.2.1.1.4.1

Factor $2$ out of $6$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{2 (3)}{16}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.4.2

Cancel the common factors.

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Step 1.2.5.2.1.1.4.2.1

Factor $2$ out of $16$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{2 \cdot 3}{2 \cdot 8}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.4.2.2

Cancel the common factor.

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{2 \cdot 3}{2 \cdot 8}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.1.4.2.3

Rewrite the expression.

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{3}{8}}{4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.2

Combine $4$ and $\frac{\sqrt{2}}{8}$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{4 \sqrt{2}}{8}}$

Step 1.2.5.2.1.3

Reduce the expression $\frac{4 \sqrt{2}}{8}$ by cancelling the common factors.

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Step 1.2.5.2.1.3.1

Factor $4$ out of $4 \sqrt{2}$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{4 (\sqrt{2})}{8}}$

Step 1.2.5.2.1.3.2

Factor $4$ out of $8$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{4 \sqrt{2}}{4 \cdot 2}}$

Step 1.2.5.2.1.3.3

Cancel the common factor.

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{4 \sqrt{2}}{4 \cdot 2}}$

Step 1.2.5.2.1.3.4

Rewrite the expression.

$e = \frac{11 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{\sqrt{2}}{2}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{11 \sqrt{2}}{8} - (\frac{3}{8} \cdot \frac{2}{\sqrt{2}})$

Step 1.2.5.2.1.5

Cancel the common factor of $2$ .

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Step 1.2.5.2.1.5.1

Factor $2$ out of $8$ .

$e = \frac{11 \sqrt{2}}{8} - (\frac{3}{2 (4)} \cdot \frac{2}{\sqrt{2}})$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = \frac{11 \sqrt{2}}{8} - (\frac{3}{2 \cdot 4} \cdot \frac{2}{\sqrt{2}})$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = \frac{11 \sqrt{2}}{8} - (\frac{3}{4} \cdot \frac{1}{\sqrt{2}})$

Step 1.2.5.2.1.6

Multiply $\frac{3}{4}$ by $\frac{1}{\sqrt{2}}$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{3}{4 \sqrt{2}}$

Step 1.2.5.2.1.7

Multiply $\frac{3}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$e = \frac{11 \sqrt{2}}{8} - (\frac{3}{4 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.2.5.2.1.8

Combine and simplify the denominator.

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Step 1.2.5.2.1.8.1

Multiply $\frac{3}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 \sqrt{2} \sqrt{2}}$

Step 1.2.5.2.1.8.2

Move $\sqrt{2}$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 (\sqrt{2} \sqrt{2})}$

Step 1.2.5.2.1.8.3

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

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 1.2.5.2.1.8.4

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

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 1.2.5.2.1.8.5

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

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 {\sqrt{2}}^{1 + 1}}$

Step 1.2.5.2.1.8.6

Add $1$ and $1$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 {\sqrt{2}}^{2}}$

Step 1.2.5.2.1.8.7

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

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Step 1.2.5.2.1.8.7.1

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

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 {(2^{\frac{1}{2}})}^{2}}$

Step 1.2.5.2.1.8.7.2

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

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.2.1.8.7.3

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

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Step 1.2.5.2.1.8.7.4

Cancel the common factor of $2$ .

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Step 1.2.5.2.1.8.7.4.1

Cancel the common factor.

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Step 1.2.5.2.1.8.7.4.2

Rewrite the expression.

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 \cdot 2^{1}}$

Step 1.2.5.2.1.8.7.5

Evaluate the exponent.

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{4 \cdot 2}$

Step 1.2.5.2.1.9

Multiply $4$ by $2$ .

$e = \frac{11 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{8}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{11 \sqrt{2} - 3 \sqrt{2}}{8}$

Step 1.2.5.2.3

Subtract $3 \sqrt{2}$ from $11 \sqrt{2}$ .

$e = \frac{8 \sqrt{2}}{8}$

Step 1.2.5.2.4

Cancel the common factor of $8$ .

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Step 1.2.5.2.4.1

Cancel the common factor.

$e = \frac{8 \sqrt{2}}{8}$

Step 1.2.5.2.4.2

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

$e = \sqrt{2}$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{\sqrt{2}}{8} {(y + \sqrt{3})}^{2} + \sqrt{2}$ .

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

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

$h = \sqrt{2}$

$k = - \sqrt{3}$

Step 3

Find the vertex $(h, k)$ .

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

Step 4