Find the Focus x^2+2x+2y+7=0

$x^{2} + 2 x + 2 y + 7 = 0$

Step 1

Rewrite the equation in vertex form.

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

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

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

$2 x + 2 y + 7 = - x^{2}$

Subtract $2 x$ from both sides of the equation.

$2 y + 7 = - x^{2} - 2 x$

Subtract $7$ from both sides of the equation.

$2 y = - x^{2} - 2 x - 7$

Divide each term in $2 y = - x^{2} - 2 x - 7$ by $2$ and simplify.

Divide each term in $2 y = - x^{2} - 2 x - 7$ by $2$ .

$\frac{2 y}{2} = \frac{- x^{2}}{2} + \frac{- 2 x}{2} + \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{- x^{2}}{2} + \frac{- 2 x}{2} + \frac{- 7}{2}$

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{2} + \frac{- 2 x}{2} + \frac{- 7}{2}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{2} + \frac{- 2 x}{2} + \frac{- 7}{2}$

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

Factor $2$ out of $- 2 x$ .

$y = - \frac{x^{2}}{2} + \frac{2 (- x)}{2} + \frac{- 7}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$y = - \frac{x^{2}}{2} + \frac{2 (- x)}{2 (1)} + \frac{- 7}{2}$

Cancel the common factor.

$y = - \frac{x^{2}}{2} + \frac{2 (- x)}{2 \cdot 1} + \frac{- 7}{2}$

Rewrite the expression.

$y = - \frac{x^{2}}{2} + \frac{- x}{1} + \frac{- 7}{2}$

Divide $- x$ by $1$ .

$y = - \frac{x^{2}}{2} - x + \frac{- 7}{2}$

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{2} - x - \frac{7}{2}$

Complete the square for $- \frac{x^{2}}{2} - x - \frac{7}{2}$ .

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

$a = - \frac{1}{2}$

$b = - 1$

$c = - \frac{7}{2}$

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

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

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

$d = \frac{1}{\frac{2}{2}}$

Divide $2$ by $2$ .

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

Divide $1$ by $1$ .

$d = 1$

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

$e = - \frac{7}{2} - \frac{{(- 1)}^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

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

$e = - \frac{7}{2} - \frac{1}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{7}{2} - \frac{1}{- 4 (\frac{1}{2})}$

Combine $- 4$ and $\frac{1}{2}$ .

$e = - \frac{7}{2} - \frac{1}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = - \frac{7}{2} - \frac{1}{- 2}$

Move the negative in front of the fraction.

$e = - \frac{7}{2} - - \frac{1}{2}$

Multiply $- - \frac{1}{2}$ .

Multiply $- 1$ by $- 1$ .

$e = - \frac{7}{2} + 1 (\frac{1}{2})$

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

$e = - \frac{7}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$e = \frac{- 7 + 1}{2}$

Add $- 7$ and $1$ .

$e = \frac{- 6}{2}$

Divide $- 6$ by $2$ .

$e = - 3$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

$a = - \frac{1}{2}$

$h = - 1$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(- 1, - 3)$

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.

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

Simplify.

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

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

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

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Combine $4$ and $\frac{1}{2}$ .

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

Divide $4$ by $2$ .

$- \frac{1}{2}$

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.

$(- 1, - \frac{7}{2})$

Step 6

Find the Focus x^2+2x+2y+7=0

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