Write in Standard Form 2y^2-3x^2=7-18x-20y

$2 y^{2} - 3 x^{2} = 7 - 18 x - 20 y$

Step 1

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

Add $18 x$ to both sides of the equation.

$2 y^{2} - 3 x^{2} + 18 x = 7 - 20 y$

Add $20 y$ to both sides of the equation.

$2 y^{2} - 3 x^{2} + 18 x + 20 y = 7$

Reorder $2 y^{2}$ and $- 3 x^{2}$ .

$- 3 x^{2} + 2 y^{2} + 18 x + 20 y = 7$

Step 2

Complete the square for $- 3 x^{2} + 18 x$ .

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

$a = - 3$

$b = 18$

$c = 0$

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

$d = \frac{18}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $18$ and $2$ .

Factor $2$ out of $18$ .

$d = \frac{2 \cdot 9}{2 \cdot - 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 3$ .

$d = \frac{2 \cdot 9}{2 (- 3)}$

Cancel the common factor.

$d = \frac{2 \cdot 9}{2 \cdot - 3}$

Rewrite the expression.

$d = \frac{9}{- 3}$

Cancel the common factor of $9$ and $- 3$ .

Factor $3$ out of $9$ .

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

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

$d = - 1 \cdot 3$

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

$e = 0 - \frac{18^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

Raise $18$ to the power of $2$ .

$e = 0 - \frac{324}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = 0 - \frac{324}{- 12}$

Divide $324$ by $- 12$ .

$e = 0 - - 27$

Multiply $- 1$ by $- 27$ .

$e = 0 + 27$

Add $0$ and $27$ .

$e = 27$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - 3)}^{2} + 27$ .

$- 3 {(x - 3)}^{2} + 27$

Step 3

Substitute $- 3 {(x - 3)}^{2} + 27$ for $- 3 x^{2} + 18 x$ in the equation $- 3 x^{2} + 2 y^{2} + 18 x + 20 y = 7$ .

$- 3 {(x - 3)}^{2} + 27 + 2 y^{2} + 20 y = 7$

Step 4

Move $27$ to the right side of the equation by adding $27$ to both sides.

$- 3 {(x - 3)}^{2} + 2 y^{2} + 20 y = 7 - 27$

Step 5

Complete the square for $2 y^{2} + 20 y$ .

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

$a = 2$

$b = 20$

$c = 0$

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

$d = \frac{20}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $20$ and $2$ .

Factor $2$ out of $20$ .

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

Cancel the common factors.

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

$d = \frac{2 \cdot 10}{2 (2)}$

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $2$ out of $10$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

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

$e = 0 - \frac{20^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $20$ to the power of $2$ .

$e = 0 - \frac{400}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 0 - \frac{400}{8}$

Divide $400$ by $8$ .

$e = 0 - 1 \cdot 50$

Multiply $- 1$ by $50$ .

$e = 0 - 50$

Subtract $50$ from $0$ .

$e = - 50$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(y + 5)}^{2} - 50$ .

$2 {(y + 5)}^{2} - 50$

Step 6

Substitute $2 {(y + 5)}^{2} - 50$ for $2 y^{2} + 20 y$ in the equation $- 3 x^{2} + 2 y^{2} + 18 x + 20 y = 7$ .

$- 3 {(x - 3)}^{2} + 2 {(y + 5)}^{2} - 50 = 7 - 27$

Step 7

Move $- 50$ to the right side of the equation by adding $50$ to both sides.

$- 3 {(x - 3)}^{2} + 2 {(y + 5)}^{2} = 7 - 27 + 50$

Step 8

Simplify $7 - 27 + 50$ .

Subtract $27$ from $7$ .

$- 3 {(x - 3)}^{2} + 2 {(y + 5)}^{2} = - 20 + 50$

Add $- 20$ and $50$ .

$- 3 {(x - 3)}^{2} + 2 {(y + 5)}^{2} = 30$

Step 9

Divide each term by $30$ to make the right side equal to one.

$- \frac{3 {(x - 3)}^{2}}{30} + \frac{2 {(y + 5)}^{2}}{30} = \frac{30}{30}$

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 $1$ .

$\frac{{(y + 5)}^{2}}{15} - \frac{{(x - 3)}^{2}}{10} = 1$

Write in Standard Form 2y^2-3x^2=7-18x-20y

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