Find the Directrix (y+3)^2=-24(x+4)

${(y + 3)}^{2} = - 24 (x + 4)$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $- 24 (x + 4) = {(y + 3)}^{2}$ .

$- 24 (x + 4) = {(y + 3)}^{2}$

Divide each term in $- 24 (x + 4) = {(y + 3)}^{2}$ by $- 24$ and simplify.

Divide each term in $- 24 (x + 4) = {(y + 3)}^{2}$ by $- 24$ .

$\frac{- 24 (x + 4)}{- 24} = \frac{{(y + 3)}^{2}}{- 24}$

Simplify the left side.

Cancel the common factor of $- 24$ .

Cancel the common factor.

$\frac{- 24 (x + 4)}{- 24} = \frac{{(y + 3)}^{2}}{- 24}$

Divide $x + 4$ by $1$ .

$x + 4 = \frac{{(y + 3)}^{2}}{- 24}$

Simplify the right side.

Move the negative in front of the fraction.

$x + 4 = - \frac{{(y + 3)}^{2}}{24}$

Subtract $4$ from both sides of the equation.

$x = - \frac{{(y + 3)}^{2}}{24} - 4$

Reorder terms.

$x = - \frac{1}{24} \cdot {(y + 3)}^{2} - 4$

Complete the square for $- \frac{1}{24} \cdot {(y + 3)}^{2} - 4$ .

Simplify the expression.

Simplify each term.

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

$- \frac{1}{24} \cdot ((y + 3) (y + 3)) - 4$

Expand $(y + 3) (y + 3)$ using the FOIL Method.

Apply the distributive property.

$- \frac{1}{24} \cdot (y (y + 3) + 3 (y + 3)) - 4$

Apply the distributive property.

$- \frac{1}{24} \cdot (y \cdot y + y \cdot 3 + 3 (y + 3)) - 4$

Apply the distributive property.

$- \frac{1}{24} \cdot (y \cdot y + y \cdot 3 + 3 y + 3 \cdot 3) - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$- \frac{1}{24} \cdot (y^{2} + y \cdot 3 + 3 y + 3 \cdot 3) - 4$

Move $3$ to the left of $y$ .

$- \frac{1}{24} \cdot (y^{2} + 3 \cdot y + 3 y + 3 \cdot 3) - 4$

Multiply $3$ by $3$ .

$- \frac{1}{24} \cdot (y^{2} + 3 y + 3 y + 9) - 4$

Add $3 y$ and $3 y$ .

$- \frac{1}{24} \cdot (y^{2} + 6 y + 9) - 4$

Apply the distributive property.

$- \frac{1}{24} y^{2} - \frac{1}{24} (6 y) - \frac{1}{24} \cdot 9 - 4$

Simplify.

Combine $y^{2}$ and $\frac{1}{24}$ .

$- \frac{y^{2}}{24} - \frac{1}{24} (6 y) - \frac{1}{24} \cdot 9 - 4$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{1}{24}$ into the numerator.

$- \frac{y^{2}}{24} + \frac{- 1}{24} (6 y) - \frac{1}{24} \cdot 9 - 4$

Factor $6$ out of $24$ .

$- \frac{y^{2}}{24} + \frac{- 1}{6 (4)} (6 y) - \frac{1}{24} \cdot 9 - 4$

Factor $6$ out of $6 y$ .

$- \frac{y^{2}}{24} + \frac{- 1}{6 (4)} (6 (y)) - \frac{1}{24} \cdot 9 - 4$

Cancel the common factor.

$- \frac{y^{2}}{24} + \frac{- 1}{6 \cdot 4} (6 y) - \frac{1}{24} \cdot 9 - 4$

Rewrite the expression.

$- \frac{y^{2}}{24} + \frac{- 1}{4} y - \frac{1}{24} \cdot 9 - 4$

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

$- \frac{y^{2}}{24} + \frac{- y}{4} - \frac{1}{24} \cdot 9 - 4$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{24}$ into the numerator.

$- \frac{y^{2}}{24} + \frac{- y}{4} + \frac{- 1}{24} \cdot 9 - 4$

Factor $3$ out of $24$ .

$- \frac{y^{2}}{24} + \frac{- y}{4} + \frac{- 1}{3 (8)} \cdot 9 - 4$

Factor $3$ out of $9$ .

$- \frac{y^{2}}{24} + \frac{- y}{4} + \frac{- 1}{3 \cdot 8} \cdot (3 \cdot 3) - 4$

Cancel the common factor.

$- \frac{y^{2}}{24} + \frac{- y}{4} + \frac{- 1}{3 \cdot 8} \cdot (3 \cdot 3) - 4$

Rewrite the expression.

$- \frac{y^{2}}{24} + \frac{- y}{4} + \frac{- 1}{8} \cdot 3 - 4$

Combine $\frac{- 1}{8}$ and $3$ .

$- \frac{y^{2}}{24} + \frac{- y}{4} + \frac{- 1 \cdot 3}{8} - 4$

Multiply $- 1$ by $3$ .

$- \frac{y^{2}}{24} + \frac{- y}{4} + \frac{- 3}{8} - 4$

Simplify each term.

Move the negative in front of the fraction.

$- \frac{y^{2}}{24} - \frac{y}{4} + \frac{- 3}{8} - 4$

Move the negative in front of the fraction.

$- \frac{y^{2}}{24} - \frac{y}{4} - \frac{3}{8} - 4$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{y^{2}}{24} - \frac{y}{4} - \frac{3}{8} - 4 \cdot \frac{8}{8}$

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

$- \frac{y^{2}}{24} - \frac{y}{4} - \frac{3}{8} + \frac{- 4 \cdot 8}{8}$

Combine the numerators over the common denominator.

$- \frac{y^{2}}{24} - \frac{y}{4} + \frac{- 3 - 4 \cdot 8}{8}$

Simplify the numerator.

Multiply $- 4$ by $8$ .

$- \frac{y^{2}}{24} - \frac{y}{4} + \frac{- 3 - 32}{8}$

Subtract $32$ from $- 3$ .

$- \frac{y^{2}}{24} - \frac{y}{4} + \frac{- 35}{8}$

Move the negative in front of the fraction.

$- \frac{y^{2}}{24} - \frac{y}{4} - \frac{35}{8}$

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

$a = - \frac{1}{24}$

$b = - \frac{1}{4}$

$c = - \frac{35}{8}$

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

$d = \frac{- \frac{1}{4}}{2 (- \frac{1}{24})}$

Simplify the right side.

Dividing two negative values results in a positive value.

$d = \frac{\frac{1}{4}}{2 (\frac{1}{24})}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{4} \cdot \frac{1}{2 (\frac{1}{24})}$

Combine $2$ and $\frac{1}{24}$ .

$d = \frac{1}{4} \cdot \frac{1}{\frac{2}{24}}$

Cancel the common factor of $2$ and $24$ .

Factor $2$ out of $2$ .

$d = \frac{1}{4} \cdot \frac{1}{\frac{2 (1)}{24}}$

Cancel the common factors.

Factor $2$ out of $24$ .

$d = \frac{1}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 12}}$

Cancel the common factor.

$d = \frac{1}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 12}}$

Rewrite the expression.

$d = \frac{1}{4} \cdot \frac{1}{\frac{1}{12}}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{4} (1 \cdot 12)$

Multiply $12$ by $1$ .

$d = \frac{1}{4} \cdot 12$

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

$d = \frac{1}{4} \cdot (4 (3))$

Cancel the common factor.

$d = \frac{1}{4} \cdot (4 \cdot 3)$

Rewrite the expression.

$d = 3$

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

$e = - \frac{35}{8} - \frac{{(- \frac{1}{4})}^{2}}{4 (- \frac{1}{24})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{1}{4}$ .

$e = - \frac{35}{8} - \frac{{(- 1)}^{2} {(\frac{1}{4})}^{2}}{4 (- \frac{1}{24})}$

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

$e = - \frac{35}{8} - \frac{1 {(\frac{1}{4})}^{2}}{4 (- \frac{1}{24})}$

Apply the product rule to $\frac{1}{4}$ .

$e = - \frac{35}{8} - \frac{1 \frac{1^{2}}{4^{2}}}{4 (- \frac{1}{24})}$

One to any power is one.

$e = - \frac{35}{8} - \frac{1 \frac{1}{4^{2}}}{4 (- \frac{1}{24})}$

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

$e = - \frac{35}{8} - \frac{1 (\frac{1}{16})}{4 (- \frac{1}{24})}$

Multiply $\frac{1}{16}$ by $1$ .

$e = - \frac{35}{8} - \frac{\frac{1}{16}}{4 (- \frac{1}{24})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{35}{8} - \frac{\frac{1}{16}}{- 4 (\frac{1}{24})}$

Combine $- 4$ and $\frac{1}{24}$ .

$e = - \frac{35}{8} - \frac{\frac{1}{16}}{\frac{- 4}{24}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $24$ .

Factor $4$ out of $- 4$ .

$e = - \frac{35}{8} - \frac{\frac{1}{16}}{\frac{4 (- 1)}{24}}$

Cancel the common factors.

Factor $4$ out of $24$ .

$e = - \frac{35}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot - 1}{4 \cdot 6}}$

Cancel the common factor.

$e = - \frac{35}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot - 1}{4 \cdot 6}}$

Rewrite the expression.

$e = - \frac{35}{8} - \frac{\frac{1}{16}}{\frac{- 1}{6}}$

Move the negative in front of the fraction.

$e = - \frac{35}{8} - \frac{\frac{1}{16}}{- \frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{35}{8} - (\frac{1}{16} (- 1 \cdot 6))$

Multiply $- 1$ by $6$ .

$e = - \frac{35}{8} - (\frac{1}{16} \cdot - 6)$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$e = - \frac{35}{8} - (\frac{1}{2 (8)} \cdot - 6)$

Factor $2$ out of $- 6$ .

$e = - \frac{35}{8} - (\frac{1}{2 \cdot 8} \cdot (2 \cdot - 3))$

Cancel the common factor.

$e = - \frac{35}{8} - (\frac{1}{2 \cdot 8} \cdot (2 \cdot - 3))$

Rewrite the expression.

$e = - \frac{35}{8} - (\frac{1}{8} \cdot - 3)$

Combine $\frac{1}{8}$ and $- 3$ .

$e = - \frac{35}{8} - \frac{- 3}{8}$

Move the negative in front of the fraction.

$e = - \frac{35}{8} - - \frac{3}{8}$

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

Multiply $- 1$ by $- 1$ .

$e = - \frac{35}{8} + 1 (\frac{3}{8})$

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

$e = - \frac{35}{8} + \frac{3}{8}$

Combine the numerators over the common denominator.

$e = \frac{- 35 + 3}{8}$

Add $- 35$ and $3$ .

$e = \frac{- 32}{8}$

Divide $- 32$ by $8$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{24} {(y + 3)}^{2} - 4$ .

$- \frac{1}{24} {(y + 3)}^{2} - 4$

Set $x$ equal to the new right side.

$x = - \frac{1}{24} \cdot {(y + 3)}^{2} - 4$

Step 2

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

$a = - \frac{1}{24}$

$h = - 4$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(- 4, - 3)$

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.

$\frac{1}{4 \cdot (- \frac{1}{24})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{24})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{24})}$

Combine $4$ and $\frac{1}{24}$ .

$- \frac{1}{\frac{4}{24}}$

Cancel the common factor of $4$ and $24$ .

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{24}}$

Cancel the common factors.

Factor $4$ out of $24$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Rewrite the expression.

$- \frac{1}{\frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 6)$

Multiply $- (1 \cdot 6)$ .

Multiply $6$ by $1$ .

$- 1 \cdot 6$

Multiply $- 1$ by $6$ .

$- 6$

Step 5

Find the directrix.

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.

$x = 2$

Step 6

Find the Directrix (y+3)^2=-24(x+4)

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