Precalculus Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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.

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

Step 1.1.2.2.1.2

Rewrite the expression.

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

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.

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

Step 1.1.2.2.2.2

Divide $y + \sqrt{3}$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Multiply $\frac{{(x - \sqrt{2})}^{2}}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.1.2.3.2

Combine and simplify the denominator.

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Step 1.1.2.3.2.1

Multiply $\frac{{(x - \sqrt{2})}^{2}}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.1.2.3.2.2

Move $\sqrt{2}$ .

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

Step 1.1.2.3.2.3

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

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

Step 1.1.2.3.2.4

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

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

Step 1.1.2.3.2.5

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

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

Step 1.1.2.3.2.6

Add $1$ and $1$ .

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

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

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

Step 1.1.2.3.2.7.2

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

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

Step 1.1.2.3.2.7.3

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

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

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.

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

Step 1.1.2.3.2.7.4.2

Rewrite the expression.

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

Step 1.1.2.3.2.7.5

Evaluate the exponent.

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

Step 1.1.2.3.3

Multiply $4$ by $2$ .

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

Step 1.1.3

Subtract $\sqrt{3}$ from both sides of the equation.

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

Step 1.1.4

Reorder terms.

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

Step 1.2

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

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

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

Step 1.2.1.2

Expand $(x - \sqrt{2}) (x - \sqrt{2})$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

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

Step 1.2.1.2.2

Apply the distributive property.

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

Step 1.2.1.2.3

Apply the distributive property.

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

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.2.1.3.1.2

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 1.2.1.3.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{\sqrt{2}}{8} \cdot (x^{2} + x (- \sqrt{2}) - \sqrt{2} x + 1 \sqrt{2} \sqrt{2}) - \sqrt{3}$

Step 1.2.1.3.1.2.2

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

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

Step 1.2.1.3.1.2.3

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

$\frac{\sqrt{2}}{8} \cdot (x^{2} + x (- \sqrt{2}) - \sqrt{2} x + {\sqrt{2}}^{1} \sqrt{2}) - \sqrt{3}$

Step 1.2.1.3.1.2.4

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

$\frac{\sqrt{2}}{8} \cdot (x^{2} + x (- \sqrt{2}) - \sqrt{2} x + {\sqrt{2}}^{1} {\sqrt{2}}^{1}) - \sqrt{3}$

Step 1.2.1.3.1.2.5

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

$\frac{\sqrt{2}}{8} \cdot (x^{2} + x (- \sqrt{2}) - \sqrt{2} x + {\sqrt{2}}^{1 + 1}) - \sqrt{3}$

Step 1.2.1.3.1.2.6

Add $1$ and $1$ .

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

Step 1.2.1.3.1.3

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

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Step 1.2.1.3.1.3.1

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

$\frac{\sqrt{2}}{8} \cdot (x^{2} + x (- \sqrt{2}) - \sqrt{2} x + {(2^{\frac{1}{2}})}^{2}) - \sqrt{3}$

Step 1.2.1.3.1.3.2

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

$\frac{\sqrt{2}}{8} \cdot (x^{2} + x (- \sqrt{2}) - \sqrt{2} x + 2^{\frac{1}{2} \cdot 2}) - \sqrt{3}$

Step 1.2.1.3.1.3.3

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

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

Step 1.2.1.3.1.3.4

Cancel the common factor of $2$ .

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Step 1.2.1.3.1.3.4.1

Cancel the common factor.

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

Step 1.2.1.3.1.3.4.2

Rewrite the expression.

$\frac{\sqrt{2}}{8} \cdot (x^{2} + x (- \sqrt{2}) - \sqrt{2} x + 2^{1}) - \sqrt{3}$

Step 1.2.1.3.1.3.5

Evaluate the exponent.

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

Step 1.2.1.3.2

Reorder the factors of $- \sqrt{2} x$ .

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

Step 1.2.1.3.3

Subtract $x \sqrt{2}$ from $x (- \sqrt{2})$ .

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Step 1.2.1.3.3.1

Reorder $x$ and $- 1$ .

$\frac{\sqrt{2}}{8} \cdot (x^{2} - 1 \cdot x \sqrt{2} - x \sqrt{2} + 2) - \sqrt{3}$

Step 1.2.1.3.3.2

Subtract $x \sqrt{2}$ from $- 1 \cdot x \sqrt{2}$ .

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

Step 1.2.1.4

Apply the distributive property.

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

Step 1.2.1.5

Simplify.

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Step 1.2.1.5.1

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

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

Step 1.2.1.5.2

Cancel the common factor of $2$ .

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Step 1.2.1.5.2.1

Factor $2$ out of $8$ .

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

Step 1.2.1.5.2.2

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

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

Step 1.2.1.5.2.3

Cancel the common factor.

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

Step 1.2.1.5.2.4

Rewrite the expression.

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

Step 1.2.1.5.3

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

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

Step 1.2.1.5.4

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

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

Step 1.2.1.5.5

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

$\frac{\sqrt{2} x^{2}}{8} - \frac{{\sqrt{2}}^{1} \sqrt{2} x}{4} + \frac{\sqrt{2}}{8} \cdot 2 - \sqrt{3}$

Step 1.2.1.5.6

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

$\frac{\sqrt{2} x^{2}}{8} - \frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1} x}{4} + \frac{\sqrt{2}}{8} \cdot 2 - \sqrt{3}$

Step 1.2.1.5.7

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

$\frac{\sqrt{2} x^{2}}{8} - \frac{{\sqrt{2}}^{1 + 1} x}{4} + \frac{\sqrt{2}}{8} \cdot 2 - \sqrt{3}$

Step 1.2.1.5.8

Add $1$ and $1$ .

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

Step 1.2.1.5.9

Cancel the common factor of $2$ .

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Step 1.2.1.5.9.1

Factor $2$ out of $8$ .

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

Step 1.2.1.5.9.2

Cancel the common factor.

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

Step 1.2.1.5.9.3

Rewrite the expression.

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

Step 1.2.1.6

Simplify each term.

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Step 1.2.1.6.1

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

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Step 1.2.1.6.1.1

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

$\frac{\sqrt{2} x^{2}}{8} - \frac{{(2^{\frac{1}{2}})}^{2} x}{4} + \frac{\sqrt{2}}{4} - \sqrt{3}$

Step 1.2.1.6.1.2

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

$\frac{\sqrt{2} x^{2}}{8} - \frac{2^{\frac{1}{2} \cdot 2} x}{4} + \frac{\sqrt{2}}{4} - \sqrt{3}$

Step 1.2.1.6.1.3

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

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

Step 1.2.1.6.1.4

Cancel the common factor of $2$ .

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Step 1.2.1.6.1.4.1

Cancel the common factor.

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

Step 1.2.1.6.1.4.2

Rewrite the expression.

$\frac{\sqrt{2} x^{2}}{8} - \frac{2^{1} x}{4} + \frac{\sqrt{2}}{4} - \sqrt{3}$

Step 1.2.1.6.1.5

Evaluate the exponent.

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

Step 1.2.1.6.2

Cancel the common factor of $2$ and $4$ .

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Step 1.2.1.6.2.1

Factor $2$ out of $2 x$ .

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

Step 1.2.1.6.2.2

Cancel the common factors.

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Step 1.2.1.6.2.2.1

Factor $2$ out of $4$ .

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

Step 1.2.1.6.2.2.2

Cancel the common factor.

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

Step 1.2.1.6.2.2.3

Rewrite the expression.

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

Step 1.2.2

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

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

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

$c = - \sqrt{3}$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- \frac{1}{2}}{2 \frac{\sqrt{2}}{8}}$

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.

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

Step 1.2.4.2.2

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

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

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 $2 \sqrt{2}$ .

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

Step 1.2.4.2.3.2

Factor $2$ out of $8$ .

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

Step 1.2.4.2.3.3

Cancel the common factor.

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

Step 1.2.4.2.3.4

Rewrite the expression.

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

Step 1.2.4.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.5

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

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

Step 1.2.4.2.6

Cancel the common factor of $2$ .

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Step 1.2.4.2.6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.2.4.2.6.2

Factor $2$ out of $4$ .

$d = \frac{- 1}{2} \cdot \frac{2 (2)}{\sqrt{2}}$

Step 1.2.4.2.6.3

Cancel the common factor.

$d = \frac{- 1}{2} \cdot \frac{2 \cdot 2}{\sqrt{2}}$

Step 1.2.4.2.6.4

Rewrite the expression.

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

Step 1.2.4.2.7

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

$d = - (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.2.4.2.8

Combine and simplify the denominator.

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Step 1.2.4.2.8.1

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

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

Step 1.2.4.2.8.2

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

$d = - \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.4.2.8.3

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

$d = - \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.4.2.8.4

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

$d = - \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.4.2.8.5

Add $1$ and $1$ .

$d = - \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.4.2.8.6

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

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Step 1.2.4.2.8.6.1

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

$d = - \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.4.2.8.6.2

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

$d = - \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.4.2.8.6.3

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

$d = - \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.4.2.8.6.4

Cancel the common factor of $2$ .

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Step 1.2.4.2.8.6.4.1

Cancel the common factor.

$d = - \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.4.2.8.6.4.2

Rewrite the expression.

$d = - \frac{2 \sqrt{2}}{2^{1}}$

Step 1.2.4.2.8.6.5

Evaluate the exponent.

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

Step 1.2.4.2.9

Cancel the common factor of $2$ .

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Step 1.2.4.2.9.1

Cancel the common factor.

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

Step 1.2.4.2.9.2

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

$d = - \sqrt{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

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

Step 1.2.5.2

Simplify each term.

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Step 1.2.5.2.1

Simplify the numerator.

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Step 1.2.5.2.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 1.2.5.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 1.2.5.2.1.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.2.5.2.1.4

One to any power is one.

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

Step 1.2.5.2.1.5

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

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

Step 1.2.5.2.1.6

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

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

Step 1.2.5.2.2

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

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

Step 1.2.5.2.3

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

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Step 1.2.5.2.3.1

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

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

Step 1.2.5.2.3.2

Factor $4$ out of $8$ .

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

Step 1.2.5.2.3.3

Cancel the common factor.

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

Step 1.2.5.2.3.4

Rewrite the expression.

$e = - \sqrt{3} - \frac{\frac{1}{4}}{\frac{\sqrt{2}}{2}}$

Step 1.2.5.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.5.2.5

Cancel the common factor of $2$ .

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Step 1.2.5.2.5.1

Factor $2$ out of $4$ .

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

Step 1.2.5.2.5.2

Cancel the common factor.

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

Step 1.2.5.2.5.3

Rewrite the expression.

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

Step 1.2.5.2.6

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

$e = - \sqrt{3} - \frac{1}{2 \sqrt{2}}$

Step 1.2.5.2.7

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

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

Step 1.2.5.2.8

Combine and simplify the denominator.

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Step 1.2.5.2.8.1

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

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

Step 1.2.5.2.8.2

Move $\sqrt{2}$ .

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

Step 1.2.5.2.8.3

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

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

Step 1.2.5.2.8.4

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

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

Step 1.2.5.2.8.5

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

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

Step 1.2.5.2.8.6

Add $1$ and $1$ .

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

Step 1.2.5.2.8.7

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

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Step 1.2.5.2.8.7.1

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

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

Step 1.2.5.2.8.7.2

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

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

Step 1.2.5.2.8.7.3

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

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

Step 1.2.5.2.8.7.4

Cancel the common factor of $2$ .

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Step 1.2.5.2.8.7.4.1

Cancel the common factor.

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

Step 1.2.5.2.8.7.4.2

Rewrite the expression.

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

Step 1.2.5.2.8.7.5

Evaluate the exponent.

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

Step 1.2.5.2.9

Multiply $2$ by $2$ .

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

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{\sqrt{2}}{8} {(x - \sqrt{2})}^{2} - \sqrt{3} - \frac{\sqrt{2}}{4}$ .

$\frac{\sqrt{2}}{8} {(x - \sqrt{2})}^{2} - \sqrt{3} - \frac{\sqrt{2}}{4}$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , 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