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

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

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

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

Subtract $6 y$ from both sides of the equation.

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

Subtract $3$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

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

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

Factor $- 3$ out of $- 6 y$ .

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

Cancel the common factors.

Factor $- 3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 y$ by $1$ .

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

Divide $- 3$ by $- 3$ .

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

Complete the square for $\frac{y^{2}}{3} + 2 y + 1$ .

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

$a = \frac{1}{3}$

$b = 2$

$c = 1$

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

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

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 3$

Multiply $3$ by $1$ .

$d = 3$

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

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

Simplify the right side.

Simplify each term.

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

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

Combine $4$ and $\frac{1}{3}$ .

$e = 1 - \frac{4}{\frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$e = 1 - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$e = 1 - 3$

Subtract $3$ from $1$ .

$e = - 2$

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

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

Set $x$ equal to the new right side.

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

Step 2

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

$a = \frac{1}{3}$

$h = - 2$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(- 2, - 3)$

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.

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

Simplify.

Combine $4$ and $\frac{1}{3}$ .

$\frac{1}{\frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{3}{4}$ by $1$ .

$\frac{3}{4}$

Step 5

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.

$(- \frac{5}{4}, - 3)$

Step 6

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

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