Write in Standard Form 4x^2-y^2-6y=25

$4 x^{2} - y^{2} - 6 y = 25$

Step 1

Complete the square for $- y^{2} - 6 y$ .

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

$a = - 1$

$b = - 6$

$c = 0$

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

$d = \frac{- 6}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{- 3}{- 1}$ .

$d = - 1 \cdot - 3$

Rewrite $- 1 \cdot - 3$ as $- - 3$ .

$d = - - 3$

Multiply $- 1$ by $- 3$ .

$d = 3$

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

$e = 0 - \frac{{(- 6)}^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $- 6$ to the power of $2$ .

$e = 0 - \frac{36}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = 0 - \frac{36}{- 4}$

Divide $36$ by $- 4$ .

$e = 0 - - 9$

Multiply $- 1$ by $- 9$ .

$e = 0 + 9$

Add $0$ and $9$ .

$e = 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(y + 3)}^{2} + 9$ .

$- {(y + 3)}^{2} + 9$

Step 2

Substitute $- {(y + 3)}^{2} + 9$ for $- y^{2} - 6 y$ in the equation $4 x^{2} - y^{2} - 6 y = 25$ .

$4 x^{2} - {(y + 3)}^{2} + 9 = 25$

Step 3

Move $9$ to the right side of the equation by adding $9$ to both sides.

$4 x^{2} - {(y + 3)}^{2} = 25 - 9$

Step 4

Subtract $9$ from $25$ .

$4 x^{2} - {(y + 3)}^{2} = 16$

Step 5

Divide each term by $16$ to make the right side equal to one.

$\frac{4 x^{2}}{16} - \frac{{(y + 3)}^{2}}{16} = \frac{16}{16}$

Step 6

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 $1$ .

$\frac{x^{2}}{4} - \frac{{(y + 3)}^{2}}{16} = 1$

Write in Standard Form 4x^2-y^2-6y=25

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