Find the Focus 6x^2+3y=0

$6 x^{2} + 3 y = 0$

Step 1

Rewrite the equation in vertex form.

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

Subtract $6 x^{2}$ from both sides of the equation.

$3 y = - 6 x^{2}$

Divide each term in $3 y = - 6 x^{2}$ by $3$ and simplify.

Divide each term in $3 y = - 6 x^{2}$ by $3$ .

$\frac{3 y}{3} = \frac{- 6 x^{2}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{- 6 x^{2}}{3}$

Divide $y$ by $1$ .

$y = \frac{- 6 x^{2}}{3}$

Simplify the right side.

Cancel the common factor of $- 6$ and $3$ .

Factor $3$ out of $- 6 x^{2}$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y = \frac{- 2 x^{2}}{1}$

Divide $- 2 x^{2}$ by $1$ .

$y = - 2 x^{2}$

Complete the square for $- 2 x^{2}$ .

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

$a = - 2$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 \cdot - 2}$

Simplify the right side.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 2}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 2$ .

$d = \frac{2 (0)}{2 (- 2)}$

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot - 2}$

Rewrite the expression.

$d = \frac{0}{- 2}$

Cancel the common factor of $0$ and $- 2$ .

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{- 2}$

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Multiply $- 1$ by $0$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = 0 - \frac{0}{- 8}$

Divide $0$ by $- 8$ .

$e = 0 - 0$

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Add $0$ and $0$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 x^{2}$ .

$- 2 x^{2}$

Set $y$ equal to the new right side.

$y = - 2 x^{2}$

Step 2

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

$a = - 2$

$h = 0$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(0, 0)$

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.

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

Simplify.

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(0, - \frac{1}{8})$

Step 6

Find the Focus 6x^2+3y=0

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