Precalculus Examples

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

Step 1

Simplify the left side $\frac{{(y - 3)}^{2}}{9} - \frac{{(x + 2)}^{2}}{1}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Divide ${(x + 2)}^{2}$ by $1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

Apply the distributive property.

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

Step 1.1.3.2

Apply the distributive property.

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

Step 1.1.3.3

Apply the distributive property.

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

Step 1.1.4

Simplify and combine like terms.

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Step 1.1.4.1

Simplify each term.

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Step 1.1.4.1.1

Multiply $x$ by $x$ .

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

Step 1.1.4.1.2

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

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

Step 1.1.4.1.3

Multiply $2$ by $2$ .

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

Step 1.1.4.2

Add $2 x$ and $2 x$ .

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

Step 1.1.5

Apply the distributive property.

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

Step 1.1.6

Simplify.

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Step 1.1.6.1

Multiply $4$ by $- 1$ .

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

Step 1.1.6.2

Multiply $- 1$ by $4$ .

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

Step 1.2

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

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

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $- x^{2}$ and $\frac{9}{9}$ .

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

Step 1.3.2

Combine the numerators over the common denominator.

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

Step 1.4

Simplify the numerator.

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Step 1.4.1

Rewrite $x^{2} \cdot 9$ as ${(x \cdot 3)}^{2}$ .

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

Step 1.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y - 3$ and $b = x \cdot 3$ .

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

Step 1.4.3

Simplify.

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Step 1.4.3.1

Move $3$ to the left of $x$ .

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

Step 1.4.3.2

Move $3$ to the left of $x$ .

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

Step 1.4.3.3

Multiply $3$ by $- 1$ .

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

Step 1.5

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

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

Step 1.6

Simplify terms.

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Step 1.6.1

Combine $- 4 x$ and $\frac{9}{9}$ .

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

Step 1.6.2

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Expand $(y - 3 + 3 x) (y - 3 - 3 x)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.7.2

Combine the opposite terms in $y \cdot y + y \cdot - 3 + y (- 3 x) - 3 y - 3 \cdot - 3 - 3 (- 3 x) + 3 x y + 3 x \cdot - 3 + 3 x (- 3 x)$ .

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Step 1.7.2.1

Reorder the factors in the terms $y (- 3 x)$ and $3 x y$ .

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

Step 1.7.2.2

Add $- 3 x y$ and $3 x y$ .

$\frac{y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3 - 3 (- 3 x) + 0 + 3 x \cdot - 3 + 3 x (- 3 x) - 4 x \cdot 9}{9} - 4 = 1$

Step 1.7.2.3

Add $y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3 - 3 (- 3 x)$ and $0$ .

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

Step 1.7.3

Simplify each term.

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Step 1.7.3.1

Multiply $y$ by $y$ .

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

Step 1.7.3.2

Move $- 3$ to the left of $y$ .

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

Step 1.7.3.3

Multiply $- 3$ by $- 3$ .

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

Step 1.7.3.4

Multiply $- 3$ by $- 3$ .

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

Step 1.7.3.5

Multiply $- 3$ by $3$ .

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

Step 1.7.3.6

Rewrite using the commutative property of multiplication.

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

Step 1.7.3.7

Multiply $x$ by $x$ by adding the exponents.

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Step 1.7.3.7.1

Move $x$ .

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

Step 1.7.3.7.2

Multiply $x$ by $x$ .

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

Step 1.7.3.8

Multiply $3$ by $- 3$ .

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

Step 1.7.4

Combine the opposite terms in $y^{2} - 3 y - 3 y + 9 + 9 x - 9 x - 9 x^{2}$ .

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Step 1.7.4.1

Subtract $9 x$ from $9 x$ .

$\frac{y^{2} - 3 y - 3 y + 9 + 0 - 9 x^{2} - 4 x \cdot 9}{9} - 4 = 1$

Step 1.7.4.2

Add $y^{2} - 3 y - 3 y + 9$ and $0$ .

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

Step 1.7.5

Subtract $3 y$ from $- 3 y$ .

$\frac{y^{2} - 6 y + 9 - 9 x^{2} - 4 x \cdot 9}{9} - 4 = 1$

Step 1.7.6

Multiply $9$ by $- 4$ .

$\frac{y^{2} - 6 y + 9 - 9 x^{2} - 36 x}{9} - 4 = 1$

Step 1.8

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

$\frac{y^{2} - 6 y + 9 - 9 x^{2} - 36 x}{9} - 4 \cdot \frac{9}{9} = 1$

Step 1.9

Simplify terms.

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Step 1.9.1

Combine $- 4$ and $\frac{9}{9}$ .

$\frac{y^{2} - 6 y + 9 - 9 x^{2} - 36 x}{9} + \frac{- 4 \cdot 9}{9} = 1$

Step 1.9.2

Combine the numerators over the common denominator.

$\frac{y^{2} - 6 y + 9 - 9 x^{2} - 36 x - 4 \cdot 9}{9} = 1$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Multiply $- 4$ by $9$ .

$\frac{y^{2} - 6 y + 9 - 9 x^{2} - 36 x - 36}{9} = 1$

Step 1.10.2

Subtract $36$ from $9$ .

$\frac{y^{2} - 6 y - 9 x^{2} - 36 x - 27}{9} = 1$

Step 2

Multiply both sides by $9$ .

$\frac{y^{2} - 6 y - 9 x^{2} - 36 x - 27}{9} \cdot 9 = 1 \cdot 9$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{y^{2} - 6 y - 9 x^{2} - 36 x - 27}{9} \cdot 9$ .

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Step 3.1.1.1

Cancel the common factor of $9$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{y^{2} - 6 y - 9 x^{2} - 36 x - 27}{9} \cdot 9 = 1 \cdot 9$

Step 3.1.1.1.2

Rewrite the expression.

$y^{2} - 6 y - 9 x^{2} - 36 x - 27 = 1 \cdot 9$

Step 3.1.1.2

Simplify the expression.

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Step 3.1.1.2.1

Move $- 6 y$ .

$y^{2} - 9 x^{2} - 36 x - 6 y - 27 = 1 \cdot 9$

Step 3.1.1.2.2

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

$- 9 x^{2} + y^{2} - 36 x - 6 y - 27 = 1 \cdot 9$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $9$ by $1$ .

$- 9 x^{2} + y^{2} - 36 x - 6 y - 27 = 9$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract $9$ from both sides of the equation.

$- 9 x^{2} + y^{2} - 36 x - 6 y - 27 - 9 = 0$

Step 4.2

Subtract $9$ from $- 27$ .

$- 9 x^{2} + y^{2} - 36 x - 6 y - 36 = 0$