Write in Standard Form 4x^2+y^2+10y+13=0

$4 x^{2} + y^{2} + 10 y + 13 = 0$

Subtract $13$ from both sides of the equation.

$4 x^{2} + y^{2} + 10 y = - 13$

Complete the square for $y^{2} + 10 y$ .

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

$a = 1$

$b = 10$

$c = 0$

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

$d = \frac{10}{2 \cdot 1}$

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

$d = \frac{2 \cdot 5}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 5}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 5}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{5}{1}$

Divide $5$ by $1$ .

$d = 5$

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

$e = 0 - \frac{10^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{100}{4}$

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Subtract $25$ from $0$ .

$e = - 25$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 5)}^{2} - 25$ .

${(y + 5)}^{2} - 25$

Substitute ${(y + 5)}^{2} - 25$ for $y^{2} + 10 y$ in the equation $4 x^{2} + y^{2} + 10 y = - 13$ .

$4 x^{2} + {(y + 5)}^{2} - 25 = - 13$

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

$4 x^{2} + {(y + 5)}^{2} = - 13 + 25$

Add $- 13$ and $25$ .

$4 x^{2} + {(y + 5)}^{2} = 12$

Divide each term by $12$ to make the right side equal to one.

$\frac{4 x^{2}}{12} + \frac{{(y + 5)}^{2}}{12} = \frac{12}{12}$

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{x^{2}}{3} + \frac{{(y + 5)}^{2}}{12} = 1$

Write in Standard Form 4x^2+y^2+10y+13=0

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