Find the Focus (x+5)^2=-30(y-6)

${(x + 5)}^{2} = - 30 (y - 6)$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $- 30 (y - 6) = {(x + 5)}^{2}$ .

$- 30 (y - 6) = {(x + 5)}^{2}$

Divide each term in $- 30 (y - 6) = {(x + 5)}^{2}$ by $- 30$ and simplify.

Divide each term in $- 30 (y - 6) = {(x + 5)}^{2}$ by $- 30$ .

$\frac{- 30 (y - 6)}{- 30} = \frac{{(x + 5)}^{2}}{- 30}$

Simplify the left side.

Cancel the common factor of $- 30$ .

Cancel the common factor.

$\frac{- 30 (y - 6)}{- 30} = \frac{{(x + 5)}^{2}}{- 30}$

Divide $y - 6$ by $1$ .

$y - 6 = \frac{{(x + 5)}^{2}}{- 30}$

Simplify the right side.

Move the negative in front of the fraction.

$y - 6 = - \frac{{(x + 5)}^{2}}{30}$

Add $6$ to both sides of the equation.

$y = - \frac{{(x + 5)}^{2}}{30} + 6$

Reorder terms.

$y = - \frac{1}{30} \cdot {(x + 5)}^{2} + 6$

Complete the square for $- \frac{1}{30} \cdot {(x + 5)}^{2} + 6$ .

Simplify the expression.

Simplify each term.

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$- \frac{1}{30} \cdot ((x + 5) (x + 5)) + 6$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$- \frac{1}{30} \cdot (x (x + 5) + 5 (x + 5)) + 6$

Apply the distributive property.

$- \frac{1}{30} \cdot (x \cdot x + x \cdot 5 + 5 (x + 5)) + 6$

Apply the distributive property.

$- \frac{1}{30} \cdot (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) + 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$- \frac{1}{30} \cdot (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) + 6$

Move $5$ to the left of $x$ .

$- \frac{1}{30} \cdot (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) + 6$

Multiply $5$ by $5$ .

$- \frac{1}{30} \cdot (x^{2} + 5 x + 5 x + 25) + 6$

Add $5 x$ and $5 x$ .

$- \frac{1}{30} \cdot (x^{2} + 10 x + 25) + 6$

Apply the distributive property.

$- \frac{1}{30} x^{2} - \frac{1}{30} (10 x) - \frac{1}{30} \cdot 25 + 6$

Simplify.

Combine $x^{2}$ and $\frac{1}{30}$ .

$- \frac{x^{2}}{30} - \frac{1}{30} (10 x) - \frac{1}{30} \cdot 25 + 6$

Cancel the common factor of $10$ .

Move the leading negative in $- \frac{1}{30}$ into the numerator.

$- \frac{x^{2}}{30} + \frac{- 1}{30} (10 x) - \frac{1}{30} \cdot 25 + 6$

Factor $10$ out of $30$ .

$- \frac{x^{2}}{30} + \frac{- 1}{10 (3)} (10 x) - \frac{1}{30} \cdot 25 + 6$

Factor $10$ out of $10 x$ .

$- \frac{x^{2}}{30} + \frac{- 1}{10 (3)} (10 (x)) - \frac{1}{30} \cdot 25 + 6$

Cancel the common factor.

$- \frac{x^{2}}{30} + \frac{- 1}{10 \cdot 3} (10 x) - \frac{1}{30} \cdot 25 + 6$

Rewrite the expression.

$- \frac{x^{2}}{30} + \frac{- 1}{3} x - \frac{1}{30} \cdot 25 + 6$

Combine $\frac{- 1}{3}$ and $x$ .

$- \frac{x^{2}}{30} + \frac{- x}{3} - \frac{1}{30} \cdot 25 + 6$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{1}{30}$ into the numerator.

$- \frac{x^{2}}{30} + \frac{- x}{3} + \frac{- 1}{30} \cdot 25 + 6$

Factor $5$ out of $30$ .

$- \frac{x^{2}}{30} + \frac{- x}{3} + \frac{- 1}{5 (6)} \cdot 25 + 6$

Factor $5$ out of $25$ .

$- \frac{x^{2}}{30} + \frac{- x}{3} + \frac{- 1}{5 \cdot 6} \cdot (5 \cdot 5) + 6$

Cancel the common factor.

$- \frac{x^{2}}{30} + \frac{- x}{3} + \frac{- 1}{5 \cdot 6} \cdot (5 \cdot 5) + 6$

Rewrite the expression.

$- \frac{x^{2}}{30} + \frac{- x}{3} + \frac{- 1}{6} \cdot 5 + 6$

Combine $\frac{- 1}{6}$ and $5$ .

$- \frac{x^{2}}{30} + \frac{- x}{3} + \frac{- 1 \cdot 5}{6} + 6$

Multiply $- 1$ by $5$ .

$- \frac{x^{2}}{30} + \frac{- x}{3} + \frac{- 5}{6} + 6$

Simplify each term.

Move the negative in front of the fraction.

$- \frac{x^{2}}{30} - \frac{x}{3} + \frac{- 5}{6} + 6$

Move the negative in front of the fraction.

$- \frac{x^{2}}{30} - \frac{x}{3} - \frac{5}{6} + 6$

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

$- \frac{x^{2}}{30} - \frac{x}{3} - \frac{5}{6} + 6 \cdot \frac{6}{6}$

Combine $6$ and $\frac{6}{6}$ .

$- \frac{x^{2}}{30} - \frac{x}{3} - \frac{5}{6} + \frac{6 \cdot 6}{6}$

Combine the numerators over the common denominator.

$- \frac{x^{2}}{30} - \frac{x}{3} + \frac{- 5 + 6 \cdot 6}{6}$

Simplify the numerator.

Multiply $6$ by $6$ .

$- \frac{x^{2}}{30} - \frac{x}{3} + \frac{- 5 + 36}{6}$

Add $- 5$ and $36$ .

$- \frac{x^{2}}{30} - \frac{x}{3} + \frac{31}{6}$

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

$a = - \frac{1}{30}$

$b = - \frac{1}{3}$

$c = \frac{31}{6}$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{- \frac{1}{3}}{2 (- \frac{1}{30})}$

Simplify the right side.

Dividing two negative values results in a positive value.

$d = \frac{\frac{1}{3}}{2 (\frac{1}{30})}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{3} \cdot \frac{1}{2 (\frac{1}{30})}$

Combine $2$ and $\frac{1}{30}$ .

$d = \frac{1}{3} \cdot \frac{1}{\frac{2}{30}}$

Cancel the common factor of $2$ and $30$ .

Factor $2$ out of $2$ .

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 (1)}{30}}$

Cancel the common factors.

Factor $2$ out of $30$ .

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 15}}$

Cancel the common factor.

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 15}}$

Rewrite the expression.

$d = \frac{1}{3} \cdot \frac{1}{\frac{1}{15}}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{3} (1 \cdot 15)$

Multiply $15$ by $1$ .

$d = \frac{1}{3} \cdot 15$

Cancel the common factor of $3$ .

Factor $3$ out of $15$ .

$d = \frac{1}{3} \cdot (3 (5))$

Cancel the common factor.

$d = \frac{1}{3} \cdot (3 \cdot 5)$

Rewrite the expression.

$d = 5$

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

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

$e = \frac{31}{6} - \frac{{(- \frac{1}{3})}^{2}}{4 (- \frac{1}{30})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{1}{3}$ .

$e = \frac{31}{6} - \frac{{(- 1)}^{2} {(\frac{1}{3})}^{2}}{4 (- \frac{1}{30})}$

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

$e = \frac{31}{6} - \frac{1 {(\frac{1}{3})}^{2}}{4 (- \frac{1}{30})}$

Apply the product rule to $\frac{1}{3}$ .

$e = \frac{31}{6} - \frac{1 \frac{1^{2}}{3^{2}}}{4 (- \frac{1}{30})}$

One to any power is one.

$e = \frac{31}{6} - \frac{1 \frac{1}{3^{2}}}{4 (- \frac{1}{30})}$

Raise $3$ to the power of $2$ .

$e = \frac{31}{6} - \frac{1 (\frac{1}{9})}{4 (- \frac{1}{30})}$

Multiply $\frac{1}{9}$ by $1$ .

$e = \frac{31}{6} - \frac{\frac{1}{9}}{4 (- \frac{1}{30})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{31}{6} - \frac{\frac{1}{9}}{- 4 (\frac{1}{30})}$

Combine $- 4$ and $\frac{1}{30}$ .

$e = \frac{31}{6} - \frac{\frac{1}{9}}{\frac{- 4}{30}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $30$ .

Factor $2$ out of $- 4$ .

$e = \frac{31}{6} - \frac{\frac{1}{9}}{\frac{2 (- 2)}{30}}$

Cancel the common factors.

Factor $2$ out of $30$ .

$e = \frac{31}{6} - \frac{\frac{1}{9}}{\frac{2 \cdot - 2}{2 \cdot 15}}$

Cancel the common factor.

$e = \frac{31}{6} - \frac{\frac{1}{9}}{\frac{2 \cdot - 2}{2 \cdot 15}}$

Rewrite the expression.

$e = \frac{31}{6} - \frac{\frac{1}{9}}{\frac{- 2}{15}}$

Move the negative in front of the fraction.

$e = \frac{31}{6} - \frac{\frac{1}{9}}{- \frac{2}{15}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{31}{6} - (\frac{1}{9} (- \frac{15}{2}))$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{15}{2}$ into the numerator.

$e = \frac{31}{6} - (\frac{1}{9} \cdot \frac{- 15}{2})$

Factor $3$ out of $9$ .

$e = \frac{31}{6} - (\frac{1}{3 (3)} \cdot \frac{- 15}{2})$

Factor $3$ out of $- 15$ .

$e = \frac{31}{6} - (\frac{1}{3 \cdot 3} \cdot \frac{3 \cdot - 5}{2})$

Cancel the common factor.

$e = \frac{31}{6} - (\frac{1}{3 \cdot 3} \cdot \frac{3 \cdot - 5}{2})$

Rewrite the expression.

$e = \frac{31}{6} - (\frac{1}{3} \cdot \frac{- 5}{2})$

Multiply $\frac{1}{3}$ by $\frac{- 5}{2}$ .

$e = \frac{31}{6} - \frac{- 5}{3 \cdot 2}$

Multiply $3$ by $2$ .

$e = \frac{31}{6} - \frac{- 5}{6}$

Move the negative in front of the fraction.

$e = \frac{31}{6} - - \frac{5}{6}$

Multiply $- - \frac{5}{6}$ .

Multiply $- 1$ by $- 1$ .

$e = \frac{31}{6} + 1 (\frac{5}{6})$

Multiply $\frac{5}{6}$ by $1$ .

$e = \frac{31}{6} + \frac{5}{6}$

Combine the numerators over the common denominator.

$e = \frac{31 + 5}{6}$

Add $31$ and $5$ .

$e = \frac{36}{6}$

Divide $36$ by $6$ .

$e = 6$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{30} {(x + 5)}^{2} + 6$ .

$- \frac{1}{30} {(x + 5)}^{2} + 6$

Set $y$ equal to the new right side.

$y = - \frac{1}{30} \cdot {(x + 5)}^{2} + 6$

Step 2

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

$a = - \frac{1}{30}$

$h = - 5$

$k = 6$

Step 3

Find the vertex $(h, k)$ .

$(- 5, 6)$

Step 4

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

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.

$\frac{1}{4 \cdot (- \frac{1}{30})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{30})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{30})}$

Combine $4$ and $\frac{1}{30}$ .

$- \frac{1}{\frac{4}{30}}$

Cancel the common factor of $4$ and $30$ .

Factor $2$ out of $4$ .

$- \frac{1}{\frac{2 (2)}{30}}$

Cancel the common factors.

Factor $2$ out of $30$ .

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 15}}$

Cancel the common factor.

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 15}}$

Rewrite the expression.

$- \frac{1}{\frac{2}{15}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{15}{2}))$

Multiply $\frac{15}{2}$ by $1$ .

$- \frac{15}{2}$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

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

$(- 5, - \frac{3}{2})$

Step 6

Find the Focus (x+5)^2=-30(y-6)

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