Trigonometry Examples

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

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

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

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

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 $\sqrt{2}$ .

Tap for more steps...

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.

Tap for more steps...

Step 1.1.2.3.1

Move the negative in front of the fraction.

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

Step 1.1.2.3.2

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 1.1.2.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 1.1.2.3.3.1

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

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

Step 1.1.2.3.3.2

Move $\sqrt{2}$ .

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

Step 1.1.2.3.3.3

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 1.1.2.3.3.4

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 1.1.2.3.3.5

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 1.1.2.3.3.6

Add $1$ and $1$ .

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

Step 1.1.2.3.3.7

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

Tap for more steps...

Step 1.1.2.3.3.7.1

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 1.1.2.3.3.7.2

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 1.1.2.3.3.7.3

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

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

Step 1.1.2.3.3.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.2.3.3.7.4.1

Cancel the common factor.

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

Step 1.1.2.3.3.7.4.2

Rewrite the expression.

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

Step 1.1.2.3.3.7.5

Evaluate the exponent.

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

Step 1.1.2.3.4

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

Tap for more steps...

Step 1.2.1

Simplify the expression.

Tap for more steps...

Step 1.2.1.1

Simplify each term.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 1.2.1.1.3.1

Simplify each term.

Tap for more steps...

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.

Tap for more steps...

Step 1.2.1.1.5.1

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

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

Step 1.2.1.1.5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.1.1.5.2.1

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

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

Step 1.2.1.1.5.2.2

Factor $2$ out of $8$ .

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

Step 1.2.1.1.5.2.3

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

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

Step 1.2.1.1.5.2.4

Cancel the common factor.

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

Step 1.2.1.1.5.2.5

Rewrite the expression.

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

Step 1.2.1.1.5.3

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

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

Step 1.2.1.1.5.4

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

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

Step 1.2.1.1.5.5

Combine using the product rule for radicals.

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

Step 1.2.1.1.5.6

Multiply $3$ by $2$ .

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

Step 1.2.1.1.5.7

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

Tap for more steps...

Step 1.2.1.1.5.7.1

Multiply $3$ by $- 1$ .

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

Step 1.2.1.1.5.7.2

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

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

Step 1.2.1.1.6

Simplify each term.

Tap for more steps...

Step 1.2.1.1.6.1

Simplify the numerator.

Tap for more steps...

Step 1.2.1.1.6.1.1

Rewrite.

$- \frac{y^{2} \sqrt{2}}{8} + \frac{0 + 0 + y (- \sqrt{6})}{4} + \frac{- 3 \sqrt{2}}{8} + \sqrt{2}$

Step 1.2.1.1.6.1.2

Add $0$ and $0$ .

$- \frac{y^{2} \sqrt{2}}{8} + \frac{0 + y (- \sqrt{6})}{4} + \frac{- 3 \sqrt{2}}{8} + \sqrt{2}$

Step 1.2.1.1.6.1.3

Remove unnecessary parentheses.

$- \frac{y^{2} \sqrt{2}}{8} + \frac{y \cdot - 1 \sqrt{6}}{4} + \frac{- 3 \sqrt{2}}{8} + \sqrt{2}$

Step 1.2.1.1.6.1.4

Factor out negative.

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

Step 1.2.1.1.6.2

Move the negative in front of the fraction.

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

Step 1.2.1.1.6.3

Move the negative in front of the fraction.

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Combine the numerators over the common denominator.

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

Step 1.2.1.5

Combine the numerators over the common denominator.

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

Step 1.2.1.6

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

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

Step 1.2.1.7

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

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

Step 1.2.1.8

Move the negative in front of the fraction.

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

Step 1.2.2

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

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

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

$c = \frac{5 \sqrt{2}}{8}$

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

Tap for more steps...

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.

Tap for more steps...

Step 1.2.4.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.4.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.3

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

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

Step 1.2.4.2.4

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

Tap for more steps...

Step 1.2.4.2.4.1

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

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

Step 1.2.4.2.4.2

Factor $2$ out of $8$ .

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

Step 1.2.4.2.4.3

Cancel the common factor.

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

Step 1.2.4.2.4.4

Rewrite the expression.

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

Step 1.2.4.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.6

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

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

Step 1.2.4.2.7

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.4.2.7.1

Cancel the common factor.

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

Step 1.2.4.2.7.2

Rewrite the expression.

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

Step 1.2.4.2.8

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

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

Step 1.2.4.2.9

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

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

Step 1.2.4.2.10

Divide $6$ by $2$ .

$d = \sqrt{3}$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

$e = \frac{5 \sqrt{2}}{8} - \frac{{(- \frac{\sqrt{6}}{4})}^{2}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.1

Simplify each term.

Tap for more steps...

Step 1.2.5.2.1.1

Simplify the numerator.

Tap for more steps...

Step 1.2.5.2.1.1.1

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

$e = \frac{5 \sqrt{2}}{8} - \frac{{(- 1)}^{2} {(\frac{\sqrt{6}}{4})}^{2}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.2

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

$e = \frac{5 \sqrt{2}}{8} - \frac{1 {(\frac{\sqrt{6}}{4})}^{2}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.3

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

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{{\sqrt{6}}^{2}}{4^{2}}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.4

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

Tap for more steps...

Step 1.2.5.2.1.1.4.1

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

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{{(6^{\frac{1}{2}})}^{2}}{4^{2}}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.4.2

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

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{6^{\frac{1}{2} \cdot 2}}{4^{2}}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.4.3

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

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{6^{\frac{2}{2}}}{4^{2}}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.5.2.1.1.4.4.1

Cancel the common factor.

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{6^{\frac{2}{2}}}{4^{2}}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.4.4.2

Rewrite the expression.

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{6^{1}}{4^{2}}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.4.5

Evaluate the exponent.

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{6}{4^{2}}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.5

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

$e = \frac{5 \sqrt{2}}{8} - \frac{1 (\frac{6}{16})}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.6

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

Tap for more steps...

Step 1.2.5.2.1.1.6.1

Factor $2$ out of $6$ .

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{2 (3)}{16}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.6.2

Cancel the common factors.

Tap for more steps...

Step 1.2.5.2.1.1.6.2.1

Factor $2$ out of $16$ .

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{2 \cdot 3}{2 \cdot 8}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.6.2.2

Cancel the common factor.

$e = \frac{5 \sqrt{2}}{8} - \frac{1 \frac{2 \cdot 3}{2 \cdot 8}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.6.2.3

Rewrite the expression.

$e = \frac{5 \sqrt{2}}{8} - \frac{1 (\frac{3}{8})}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.1.7

Multiply $\frac{3}{8}$ by $1$ .

$e = \frac{5 \sqrt{2}}{8} - \frac{\frac{3}{8}}{4 (- \frac{\sqrt{2}}{8})}$

Step 1.2.5.2.1.2

Simplify the denominator.

Tap for more steps...

Step 1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = \frac{5 \sqrt{2}}{8} - \frac{\frac{3}{8}}{- 4 \frac{\sqrt{2}}{8}}$

Step 1.2.5.2.1.2.2

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

$e = \frac{5 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{- 4 \sqrt{2}}{8}}$

Step 1.2.5.2.1.3

Simplify the denominator.

Tap for more steps...

Step 1.2.5.2.1.3.1

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

Tap for more steps...

Step 1.2.5.2.1.3.1.1

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

$e = \frac{5 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{4 (- \sqrt{2})}{8}}$

Step 1.2.5.2.1.3.1.2

Factor $4$ out of $8$ .

$e = \frac{5 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{4 (- \sqrt{2})}{4 (2)}}$

Step 1.2.5.2.1.3.1.3

Cancel the common factor.

$e = \frac{5 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{4 (- \sqrt{2})}{4 \cdot 2}}$

Step 1.2.5.2.1.3.1.4

Rewrite the expression.

$e = \frac{5 \sqrt{2}}{8} - \frac{\frac{3}{8}}{\frac{- \sqrt{2}}{2}}$

Step 1.2.5.2.1.3.2

Move the negative in front of the fraction.

$e = \frac{5 \sqrt{2}}{8} - \frac{\frac{3}{8}}{- \frac{\sqrt{2}}{2}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{5 \sqrt{2}}{8} - (\frac{3}{8} (- \frac{2}{\sqrt{2}}))$

Step 1.2.5.2.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.5.2.1.5.1

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

$e = \frac{5 \sqrt{2}}{8} - (\frac{3}{8} \cdot \frac{- 2}{\sqrt{2}})$

Step 1.2.5.2.1.5.2

Factor $2$ out of $8$ .

$e = \frac{5 \sqrt{2}}{8} - (\frac{3}{2 (4)} \cdot \frac{- 2}{\sqrt{2}})$

Step 1.2.5.2.1.5.3

Factor $2$ out of $- 2$ .

$e = \frac{5 \sqrt{2}}{8} - (\frac{3}{2 \cdot 4} \cdot \frac{2 \cdot - 1}{\sqrt{2}})$

Step 1.2.5.2.1.5.4

Cancel the common factor.

$e = \frac{5 \sqrt{2}}{8} - (\frac{3}{2 \cdot 4} \cdot \frac{2 \cdot - 1}{\sqrt{2}})$

Step 1.2.5.2.1.5.5

Rewrite the expression.

$e = \frac{5 \sqrt{2}}{8} - (\frac{3}{4} \cdot \frac{- 1}{\sqrt{2}})$

Step 1.2.5.2.1.6

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

$e = \frac{5 \sqrt{2}}{8} - \frac{3 \cdot - 1}{4 \sqrt{2}}$

Step 1.2.5.2.1.7

Multiply $3$ by $- 1$ .

$e = \frac{5 \sqrt{2}}{8} - \frac{- 3}{4 \sqrt{2}}$

Step 1.2.5.2.1.8

Move the negative in front of the fraction.

$e = \frac{5 \sqrt{2}}{8} - - \frac{3}{4 \sqrt{2}}$

Step 1.2.5.2.1.9

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

$e = \frac{5 \sqrt{2}}{8} - - (\frac{3}{4 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.2.5.2.1.10

Combine and simplify the denominator.

Tap for more steps...

Step 1.2.5.2.1.10.1

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

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 \sqrt{2} \sqrt{2}}$

Step 1.2.5.2.1.10.2

Move $\sqrt{2}$ .

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 (\sqrt{2} \sqrt{2})}$

Step 1.2.5.2.1.10.3

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

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 1.2.5.2.1.10.4

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

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 1.2.5.2.1.10.5

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

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 {\sqrt{2}}^{1 + 1}}$

Step 1.2.5.2.1.10.6

Add $1$ and $1$ .

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 {\sqrt{2}}^{2}}$

Step 1.2.5.2.1.10.7

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

Tap for more steps...

Step 1.2.5.2.1.10.7.1

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

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 {(2^{\frac{1}{2}})}^{2}}$

Step 1.2.5.2.1.10.7.2

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

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.2.1.10.7.3

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

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Step 1.2.5.2.1.10.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.5.2.1.10.7.4.1

Cancel the common factor.

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Step 1.2.5.2.1.10.7.4.2

Rewrite the expression.

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 \cdot 2^{1}}$

Step 1.2.5.2.1.10.7.5

Evaluate the exponent.

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{4 \cdot 2}$

Step 1.2.5.2.1.11

Multiply $4$ by $2$ .

$e = \frac{5 \sqrt{2}}{8} - - \frac{3 \sqrt{2}}{8}$

Step 1.2.5.2.1.12

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

Tap for more steps...

Step 1.2.5.2.1.12.1

Multiply $- 1$ by $- 1$ .

$e = \frac{5 \sqrt{2}}{8} + 1 \frac{3 \sqrt{2}}{8}$

Step 1.2.5.2.1.12.2

Multiply $\frac{3 \sqrt{2}}{8}$ by $1$ .

$e = \frac{5 \sqrt{2}}{8} + \frac{3 \sqrt{2}}{8}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{5 \sqrt{2} + 3 \sqrt{2}}{8}$

Step 1.2.5.2.3

Add $5 \sqrt{2}$ and $3 \sqrt{2}$ .

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

Step 1.2.5.2.4

Cancel the common factor of $8$ .

Tap for more steps...

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

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

Tap for more steps...

Step 4.1

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

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

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

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 4.3.1.1

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

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

Step 4.3.1.2

Move the negative in front of the fraction.

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

Step 4.3.2

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

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

Step 4.3.3

Cancel the common factor of $4$ and $8$ .

Tap for more steps...

Step 4.3.3.1

Factor $4$ out of $4$ .

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

Step 4.3.3.2

Cancel the common factors.

Tap for more steps...

Step 4.3.3.2.1

Factor $4$ out of $8$ .

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

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

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Step 4.3.5

Multiply $- (1 \cdot 2)$ .

Tap for more steps...

Step 4.3.5.1

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Step 4.3.5.2

Multiply $- 1$ by $2$ .

$- 2$

Step 5

Find the directrix.

Tap for more steps...

Step 5.1

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.

$x = h - p$

Step 5.2

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

$x = \sqrt{2} + 2$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2} + 2$

Decimal Form:

$x = 3.41421356 \dots$

Step 7