Find the Vertices ((x-2)^2)/9-((y-1)^2)/4

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

Subtract $\frac{{(x - 2)}^{2}}{9}$ from both sides of the equation.

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

Add $\frac{{(y - 1)}^{2}}{4}$ to both sides of the equation.

$y - \frac{{(x - 2)}^{2}}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

To write $y$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y \cdot \frac{9}{9} - \frac{{(x - 2)}^{2}}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Combine $y$ and $\frac{9}{9}$ .

$\frac{y \cdot 9}{9} - \frac{{(x - 2)}^{2}}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Combine the numerators over the common denominator.

$\frac{y \cdot 9 - {(x - 2)}^{2}}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Simplify each term.

Simplify the numerator.

Move $9$ to the left of $y$ .

$\frac{9 \cdot y - {(x - 2)}^{2}}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$\frac{9 y - ((x - 2) (x - 2))}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{9 y - (x (x - 2) - 2 (x - 2))}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Apply the distributive property.

$\frac{9 y - (x \cdot x + x \cdot - 2 - 2 (x - 2))}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Apply the distributive property.

$\frac{9 y - (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{9 y - (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Move $- 2$ to the left of $x$ .

$\frac{9 y - (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Multiply $- 2$ by $- 2$ .

$\frac{9 y - (x^{2} - 2 x - 2 x + 4)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Subtract $2 x$ from $- 2 x$ .

$\frac{9 y - (x^{2} - 4 x + 4)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Apply the distributive property.

$\frac{9 y - x^{2} - (- 4 x) - 1 \cdot 4}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Simplify.

Multiply $- 4$ by $- 1$ .

$\frac{9 y - x^{2} + 4 x - 1 \cdot 4}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Multiply $- 1$ by $4$ .

$\frac{9 y - x^{2} + 4 x - 4}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Rewrite $9 y - x^{2} + 4 x - 4$ in a factored form.

Regroup terms.

$\frac{- 4 + 9 y - x^{2} + 4 x}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Factor $- 1$ out of $9 y - x^{2} + 4 x$ .

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

$\frac{- 4 - x^{2} + 9 y + 4 x}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Factor $- 1$ out of $- x^{2}$ .

$\frac{- 4 - (x^{2}) + 9 y + 4 x}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Factor $- 1$ out of $9 y$ .

$\frac{- 4 - (x^{2}) - (- 9 y) + 4 x}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Factor $- 1$ out of $4 x$ .

$\frac{- 4 - (x^{2}) - (- 9 y) - (- 4 x)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 9 y)$ .

$\frac{- 4 - (x^{2} - 9 y) - (- 4 x)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Factor $- 1$ out of $- (x^{2} - 9 y) - (- 4 x)$ .

$\frac{- 4 - (x^{2} - 9 y - 4 x)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Factor $- 1$ out of $- 4 - (x^{2} - 9 y - 4 x)$ .

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

$\frac{- (x^{2} - 9 y - 4 x) - 4}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- (x^{2} - 9 y - 4 x) - 1 (4)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Factor $- 1$ out of $- (x^{2} - 9 y - 4 x) - 1 (4)$ .

$\frac{- (x^{2} - 9 y - 4 x + 4)}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 9 y - 4 x + 4}{9} + \frac{{(y - 1)}^{2}}{4} = 0$

To write $- \frac{x^{2} - 9 y - 4 x + 4}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{x^{2} - 9 y - 4 x + 4}{9} \cdot \frac{4}{4} + \frac{{(y - 1)}^{2}}{4} = 0$

To write $\frac{{(y - 1)}^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{x^{2} - 9 y - 4 x + 4}{9} \cdot \frac{4}{4} + \frac{{(y - 1)}^{2}}{4} \cdot \frac{9}{9} = 0$

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x^{2} - 9 y - 4 x + 4}{9}$ by $\frac{4}{4}$ .

$- \frac{(x^{2} - 9 y - 4 x + 4) \cdot 4}{9 \cdot 4} + \frac{{(y - 1)}^{2}}{4} \cdot \frac{9}{9} = 0$

Multiply $9$ by $4$ .

$- \frac{(x^{2} - 9 y - 4 x + 4) \cdot 4}{36} + \frac{{(y - 1)}^{2}}{4} \cdot \frac{9}{9} = 0$

Multiply $\frac{{(y - 1)}^{2}}{4}$ by $\frac{9}{9}$ .

$- \frac{(x^{2} - 9 y - 4 x + 4) \cdot 4}{36} + \frac{{(y - 1)}^{2} \cdot 9}{4 \cdot 9} = 0$

Multiply $4$ by $9$ .

$- \frac{(x^{2} - 9 y - 4 x + 4) \cdot 4}{36} + \frac{{(y - 1)}^{2} \cdot 9}{36} = 0$

Combine the numerators over the common denominator.

$\frac{- (x^{2} - 9 y - 4 x + 4) \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{(- x^{2} - (- 9 y) - (- 4 x) - 1 \cdot 4) \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Simplify.

Multiply $- 9$ by $- 1$ .

$\frac{(- x^{2} + 9 y - (- 4 x) - 1 \cdot 4) \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Multiply $- 4$ by $- 1$ .

$\frac{(- x^{2} + 9 y + 4 x - 1 \cdot 4) \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Multiply $- 1$ by $4$ .

$\frac{(- x^{2} + 9 y + 4 x - 4) \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Apply the distributive property.

$\frac{- x^{2} \cdot 4 + 9 y \cdot 4 + 4 x \cdot 4 - 4 \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Simplify.

Multiply $4$ by $- 1$ .

$\frac{- 4 x^{2} + 9 y \cdot 4 + 4 x \cdot 4 - 4 \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Multiply $4$ by $9$ .

$\frac{- 4 x^{2} + 36 y + 4 x \cdot 4 - 4 \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Multiply $4$ by $4$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 4 \cdot 4 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Multiply $- 4$ by $4$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + {(y - 1)}^{2} \cdot 9}{36} = 0$

Rewrite ${(y - 1)}^{2}$ as $(y - 1) (y - 1)$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y - 1) (y - 1) \cdot 9}{36} = 0$

Expand $(y - 1) (y - 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y (y - 1) - 1 (y - 1)) \cdot 9}{36} = 0$

Apply the distributive property.

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y \cdot y + y \cdot - 1 - 1 (y - 1)) \cdot 9}{36} = 0$

Apply the distributive property.

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y \cdot y + y \cdot - 1 - 1 y - 1 \cdot - 1) \cdot 9}{36} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y^{2} + y \cdot - 1 - 1 y - 1 \cdot - 1) \cdot 9}{36} = 0$

Move $- 1$ to the left of $y$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y^{2} - 1 \cdot y - 1 y - 1 \cdot - 1) \cdot 9}{36} = 0$

Rewrite $- 1 y$ as $- y$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y^{2} - y - 1 y - 1 \cdot - 1) \cdot 9}{36} = 0$

Rewrite $- 1 y$ as $- y$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y^{2} - y - y - 1 \cdot - 1) \cdot 9}{36} = 0$

Multiply $- 1$ by $- 1$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y^{2} - y - y + 1) \cdot 9}{36} = 0$

Subtract $y$ from $- y$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + (y^{2} - 2 y + 1) \cdot 9}{36} = 0$

Apply the distributive property.

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + y^{2} \cdot 9 - 2 y \cdot 9 + 1 \cdot 9}{36} = 0$

Simplify.

Move $9$ to the left of $y^{2}$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + 9 \cdot y^{2} - 2 y \cdot 9 + 1 \cdot 9}{36} = 0$

Multiply $9$ by $- 2$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + 9 \cdot y^{2} - 18 y + 1 \cdot 9}{36} = 0$

Multiply $9$ by $1$ .

$\frac{- 4 x^{2} + 36 y + 16 x - 16 + 9 \cdot y^{2} - 18 y + 9}{36} = 0$

Subtract $18 y$ from $36 y$ .

$\frac{- 4 x^{2} + 16 x - 16 + 9 y^{2} + 18 y + 9}{36} = 0$

Add $- 16$ and $9$ .

$\frac{- 4 x^{2} + 16 x + 9 y^{2} + 18 y - 7}{36} = 0$

Factor $- 1$ out of $- 4 x^{2}$ .

$\frac{- (4 x^{2}) + 16 x + 9 y^{2} + 18 y - 7}{36} = 0$

Factor $- 1$ out of $16 x$ .

$\frac{- (4 x^{2}) - (- 16 x) + 9 y^{2} + 18 y - 7}{36} = 0$

Factor $- 1$ out of $- (4 x^{2}) - (- 16 x)$ .

$\frac{- (4 x^{2} - 16 x) + 9 y^{2} + 18 y - 7}{36} = 0$

Factor $- 1$ out of $9 y^{2}$ .

$\frac{- (4 x^{2} - 16 x) - (- 9 y^{2}) + 18 y - 7}{36} = 0$

Factor $- 1$ out of $- (4 x^{2} - 16 x) - (- 9 y^{2})$ .

$\frac{- (4 x^{2} - 16 x - 9 y^{2}) + 18 y - 7}{36} = 0$

Factor $- 1$ out of $18 y$ .

$\frac{- (4 x^{2} - 16 x - 9 y^{2}) - (- 18 y) - 7}{36} = 0$

Factor $- 1$ out of $- (4 x^{2} - 16 x - 9 y^{2}) - (- 18 y)$ .

$\frac{- (4 x^{2} - 16 x - 9 y^{2} - 18 y) - 7}{36} = 0$

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{- (4 x^{2} - 16 x - 9 y^{2} - 18 y) - 1 (7)}{36} = 0$

Factor $- 1$ out of $- (4 x^{2} - 16 x - 9 y^{2} - 18 y) - 1 (7)$ .

$\frac{- (4 x^{2} - 16 x - 9 y^{2} - 18 y + 7)}{36} = 0$

Rewrite $- (4 x^{2} - 16 x - 9 y^{2} - 18 y + 7)$ as $- 1 (4 x^{2} - 16 x - 9 y^{2} - 18 y + 7)$ .

$\frac{- 1 (4 x^{2} - 16 x - 9 y^{2} - 18 y + 7)}{36} = 0$

Move the negative in front of the fraction.

$- \frac{4 x^{2} - 16 x - 9 y^{2} - 18 y + 7}{36} = 0$

Find the Vertices ((x-2)^2)/9-((y-1)^2)/4

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