Write in Standard Form 9y^2+144y+504=4x^2+24x

$9 y^{2} + 144 y + 504 = 4 x^{2} + 24 x$

Move all terms containing variables to the left side of the equation.

Subtract $4 x^{2}$ from both sides of the equation.

$9 y^{2} + 144 y + 504 - 4 x^{2} = 24 x$

Subtract $24 x$ from both sides of the equation.

$9 y^{2} + 144 y + 504 - 4 x^{2} - 24 x = 0$

Move $504$ .

$9 y^{2} + 144 y - 4 x^{2} - 24 x + 504 = 0$

Move $144 y$ .

$9 y^{2} - 4 x^{2} - 24 x + 144 y + 504 = 0$

Reorder $9 y^{2}$ and $- 4 x^{2}$ .

$- 4 x^{2} + 9 y^{2} - 24 x + 144 y + 504 = 0$

Subtract $504$ from both sides of the equation.

$- 4 x^{2} + 9 y^{2} - 24 x + 144 y = - 504$

Complete the square for $- 4 x^{2} - 24 x$ .

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

$a = - 4$

$b = - 24$

$c = 0$

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

$d = \frac{- 24}{2 \cdot - 4}$

Simplify the right side.

Cancel the common factor of $- 24$ and $2$ .

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot - 12}{2 (- 4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot - 4}$

Rewrite the expression.

$d = \frac{- 12}{- 4}$

Cancel the common factor of $- 12$ and $- 4$ .

Factor $- 4$ out of $- 12$ .

$d = \frac{- 4 (3)}{- 4}$

Cancel the common factors.

Factor $- 4$ out of $- 4$ .

$d = \frac{- 4 \cdot 3}{- 4 \cdot 1}$

Cancel the common factor.

$d = \frac{- 4 \cdot 3}{- 4 \cdot 1}$

Rewrite the expression.

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

Divide $3$ 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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 24)}^{2}}{4 \cdot - 4}$

Simplify the right side.

Simplify each term.

Raise $- 24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot - 4}$

Multiply $4$ by $- 4$ .

$e = 0 - \frac{576}{- 16}$

Divide $576$ by $- 16$ .

$e = 0 - - 36$

Multiply $- 1$ by $- 36$ .

$e = 0 + 36$

Add $0$ and $36$ .

$e = 36$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x + 3)}^{2} + 36$ .

$- 4 {(x + 3)}^{2} + 36$

Substitute $- 4 {(x + 3)}^{2} + 36$ for $- 4 x^{2} - 24 x$ in the equation $- 4 x^{2} + 9 y^{2} - 24 x + 144 y = - 504$ .

$- 4 {(x + 3)}^{2} + 36 + 9 y^{2} + 144 y = - 504$

Move $36$ to the right side of the equation by adding $36$ to both sides.

$- 4 {(x + 3)}^{2} + 9 y^{2} + 144 y = - 504 - 36$

Complete the square for $9 y^{2} + 144 y$ .

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

$a = 9$

$b = 144$

$c = 0$

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

$d = \frac{144}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $144$ and $2$ .

Factor $2$ out of $144$ .

$d = \frac{2 \cdot 72}{2 \cdot 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot 72}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot 72}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{72}{9}$

Cancel the common factor of $72$ and $9$ .

Factor $9$ out of $72$ .

$d = \frac{9 \cdot 8}{9}$

Cancel the common factors.

Factor $9$ out of $9$ .

$d = \frac{9 \cdot 8}{9 (1)}$

Cancel the common factor.

$d = \frac{9 \cdot 8}{9 \cdot 1}$

Rewrite the expression.

$d = \frac{8}{1}$

Divide $8$ by $1$ .

$d = 8$

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

$e = 0 - \frac{144^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $144$ to the power of $2$ .

$e = 0 - \frac{20736}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{20736}{36}$

Divide $20736$ by $36$ .

$e = 0 - 1 \cdot 576$

Multiply $- 1$ by $576$ .

$e = 0 - 576$

Subtract $576$ from $0$ .

$e = - 576$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(y + 8)}^{2} - 576$ .

$9 {(y + 8)}^{2} - 576$

Substitute $9 {(y + 8)}^{2} - 576$ for $9 y^{2} + 144 y$ in the equation $- 4 x^{2} + 9 y^{2} - 24 x + 144 y = - 504$ .

$- 4 {(x + 3)}^{2} + 9 {(y + 8)}^{2} - 576 = - 504 - 36$

Move $- 576$ to the right side of the equation by adding $576$ to both sides.

$- 4 {(x + 3)}^{2} + 9 {(y + 8)}^{2} = - 504 - 36 + 576$

Simplify $- 504 - 36 + 576$ .

Subtract $36$ from $- 504$ .

$- 4 {(x + 3)}^{2} + 9 {(y + 8)}^{2} = - 540 + 576$

Add $- 540$ and $576$ .

$- 4 {(x + 3)}^{2} + 9 {(y + 8)}^{2} = 36$

Divide each term by $36$ to make the right side equal to one.

$- \frac{4 {(x + 3)}^{2}}{36} + \frac{9 {(y + 8)}^{2}}{36} = \frac{36}{36}$

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 $1$ .

$\frac{{(y + 8)}^{2}}{4} - \frac{{(x + 3)}^{2}}{9} = 1$

Write in Standard Form 9y^2+144y+504=4x^2+24x

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