Find the Axis of Symmetry 3y^2+8x+24y+40=0

$3 y^{2} + 8 x + 24 y + 40 = 0$

Step 1

Rewrite the equation in vertex form.

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

Move all terms not containing $x$ to the right side of the equation.

Subtract $3 y^{2}$ from both sides of the equation.

$8 x + 24 y + 40 = - 3 y^{2}$

Subtract $24 y$ from both sides of the equation.

$8 x + 40 = - 3 y^{2} - 24 y$

Subtract $40$ from both sides of the equation.

$8 x = - 3 y^{2} - 24 y - 40$

Divide each term in $8 x = - 3 y^{2} - 24 y - 40$ by $8$ and simplify.

Divide each term in $8 x = - 3 y^{2} - 24 y - 40$ by $8$ .

$\frac{8 x}{8} = \frac{- 3 y^{2}}{8} + \frac{- 24 y}{8} + \frac{- 40}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 3 y^{2}}{8} + \frac{- 24 y}{8} + \frac{- 40}{8}$

Divide $x$ by $1$ .

$x = \frac{- 3 y^{2}}{8} + \frac{- 24 y}{8} + \frac{- 40}{8}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{(3) y^{2}}{8} + \frac{- 24 y}{8} + \frac{- 40}{8}$

Cancel the common factor of $- 24$ and $8$ .

Factor $8$ out of $- 24 y$ .

$x = - \frac{3 y^{2}}{8} + \frac{8 (- 3 y)}{8} + \frac{- 40}{8}$

Cancel the common factors.

Factor $8$ out of $8$ .

$x = - \frac{3 y^{2}}{8} + \frac{8 (- 3 y)}{8 (1)} + \frac{- 40}{8}$

Cancel the common factor.

$x = - \frac{3 y^{2}}{8} + \frac{8 (- 3 y)}{8 \cdot 1} + \frac{- 40}{8}$

Rewrite the expression.

$x = - \frac{3 y^{2}}{8} + \frac{- 3 y}{1} + \frac{- 40}{8}$

Divide $- 3 y$ by $1$ .

$x = - \frac{3 y^{2}}{8} - 3 y + \frac{- 40}{8}$

Divide $- 40$ by $8$ .

$x = - \frac{3 y^{2}}{8} - 3 y - 5$

Complete the square for $- \frac{3 y^{2}}{8} - 3 y - 5$ .

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

$a = - \frac{3}{8}$

$b = - 3$

$c = - 5$

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

$d = \frac{- 3}{2 (- \frac{3}{8})}$

Simplify the right side.

Simplify the denominator.

Multiply $- 1$ by $2$ .

$d = \frac{- 3}{- 2 (\frac{3}{8})}$

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

$d = \frac{- 3}{\frac{- 2 \cdot 3}{8}}$

Multiply $- 2$ by $3$ .

$d = \frac{- 3}{\frac{- 6}{8}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 6$ and $8$ .

Factor $2$ out of $- 6$ .

$d = \frac{- 3}{\frac{2 (- 3)}{8}}$

Cancel the common factors.

Factor $2$ out of $8$ .

$d = \frac{- 3}{\frac{2 \cdot - 3}{2 \cdot 4}}$

Cancel the common factor.

$d = \frac{- 3}{\frac{2 \cdot - 3}{2 \cdot 4}}$

Rewrite the expression.

$d = \frac{- 3}{\frac{- 3}{4}}$

Move the negative in front of the fraction.

$d = \frac{- 3}{- \frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - 3 (- \frac{4}{3})$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$d = - 3 (\frac{- 4}{3})$

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

$d = 3 \cdot - 1 \frac{- 4}{3}$

Rewrite the expression.

$d = - - 4$

Multiply $- 1$ by $- 4$ .

$d = 4$

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

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

Simplify the right side.

Simplify each term.

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

$e = - 5 - \frac{9}{4 (- \frac{3}{8})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - 5 - \frac{9}{- 4 (\frac{3}{8})}$

Combine $- 4$ and $\frac{3}{8}$ .

$e = - 5 - \frac{9}{\frac{- 4 \cdot 3}{8}}$

Multiply $- 4$ by $3$ .

$e = - 5 - \frac{9}{\frac{- 12}{8}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 12$ and $8$ .

Factor $4$ out of $- 12$ .

$e = - 5 - \frac{9}{\frac{4 (- 3)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$e = - 5 - \frac{9}{\frac{4 \cdot - 3}{4 \cdot 2}}$

Cancel the common factor.

$e = - 5 - \frac{9}{\frac{4 \cdot - 3}{4 \cdot 2}}$

Rewrite the expression.

$e = - 5 - \frac{9}{\frac{- 3}{2}}$

Move the negative in front of the fraction.

$e = - 5 - \frac{9}{- \frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - 5 - (9 (- \frac{2}{3}))$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$e = - 5 - (9 (\frac{- 2}{3}))$

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

$e = - 5 - (3 \cdot - 2)$

Multiply $3$ by $- 2$ .

$e = - 5 - - 6$

Multiply $- 1$ by $- 6$ .

$e = - 5 + 6$

Add $- 5$ and $6$ .

$e = 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{3}{8} {(y + 4)}^{2} + 1$ .

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

Set $x$ equal to the new right side.

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

Step 2

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

$a = - \frac{3}{8}$

$h = 1$

$k = - 4$

Step 3

Since the value of $a$ is negative, the parabola opens left.

Opens Left

Step 4

Find the vertex $(h, k)$ .

$(1, - 4)$

Step 5

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.

$\frac{1}{4 \cdot (- \frac{3}{8})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{3}{8})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{3}{8})}$

Combine $4$ and $\frac{3}{8}$ .

$- \frac{1}{\frac{4 \cdot 3}{8}}$

Multiply $4$ by $3$ .

$- \frac{1}{\frac{12}{8}}$

Cancel the common factor of $12$ and $8$ .

Factor $4$ out of $12$ .

$- \frac{1}{\frac{4 (3)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$- \frac{1}{\frac{4 \cdot 3}{4 \cdot 2}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 3}{4 \cdot 2}}$

Rewrite the expression.

$- \frac{1}{\frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{2}{3}))$

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

$- \frac{2}{3}$

Step 6

Find the focus.

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

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

$(\frac{1}{3}, - 4)$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$y = - 4$

Step 8

Find the Axis of Symmetry 3y^2+8x+24y+40=0

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