Write in Standard Form 9x^2+4y^2-36x+8y+4=0

$9 x^{2} + 4 y^{2} - 36 x + 8 y + 4 = 0$

Step 1

Subtract $4$ from both sides of the equation.

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

Step 2

Complete the square for $9 x^{2} - 36 x$ .

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

$a = 9$

$b = - 36$

$c = 0$

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

$d = \frac{- 36}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $- 36$ and $2$ .

Factor $2$ out of $- 36$ .

$d = \frac{2 \cdot - 18}{2 \cdot 9}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 18}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 18}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{- 18}{9}$

Cancel the common factor of $- 18$ and $9$ .

Factor $9$ out of $- 18$ .

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

Cancel the common factors.

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 2}{9 (1)}$

Cancel the common factor.

$d = \frac{9 \cdot - 2}{9 \cdot 1}$

Rewrite the expression.

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

Divide $- 2$ by $1$ .

$d = - 2$

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

$e = 0 - \frac{{(- 36)}^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $- 36$ to the power of $2$ .

$e = 0 - \frac{1296}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{1296}{36}$

Divide $1296$ by $36$ .

$e = 0 - 1 \cdot 36$

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Subtract $36$ from $0$ .

$e = - 36$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 2)}^{2} - 36$ .

$9 {(x - 2)}^{2} - 36$

Step 3

Substitute $9 {(x - 2)}^{2} - 36$ for $9 x^{2} - 36 x$ in the equation $9 x^{2} + 4 y^{2} - 36 x + 8 y = - 4$ .

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

Step 4

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

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

Step 5

Complete the square for $4 y^{2} + 8 y$ .

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

$a = 4$

$b = 8$

$c = 0$

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

$d = \frac{8}{2 \cdot 4}$

Simplify the right side.

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

$e = 0 - \frac{8^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y + 1)}^{2} - 4$ .

$4 {(y + 1)}^{2} - 4$

Step 6

Substitute $4 {(y + 1)}^{2} - 4$ for $4 y^{2} + 8 y$ in the equation $9 x^{2} + 4 y^{2} - 36 x + 8 y = - 4$ .

$9 {(x - 2)}^{2} + 4 {(y + 1)}^{2} - 4 = - 4 + 36$

Step 7

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

$9 {(x - 2)}^{2} + 4 {(y + 1)}^{2} = - 4 + 36 + 4$

Step 8

Simplify $- 4 + 36 + 4$ .

Add $- 4$ and $36$ .

$9 {(x - 2)}^{2} + 4 {(y + 1)}^{2} = 32 + 4$

Add $32$ and $4$ .

$9 {(x - 2)}^{2} + 4 {(y + 1)}^{2} = 36$

Step 9

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

$\frac{9 {(x - 2)}^{2}}{36} + \frac{4 {(y + 1)}^{2}}{36} = \frac{36}{36}$

Step 10

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 $1$ .

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

Write in Standard Form 9x^2+4y^2-36x+8y+4=0

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