Precalculus Examples

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

Step 1

Simplify the left side $\frac{{(x - 1)}^{2}}{9} - \frac{{(y - 3)}^{2}}{4}$ .

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Step 1.1

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

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

Step 1.2

To write $- \frac{{(y - 3)}^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Multiply $9$ by $4$ .

$\frac{{(x - 1)}^{2} \cdot 4}{36} - \frac{{(y - 3)}^{2}}{4} \cdot \frac{9}{9} = 1$

Step 1.3.3

Multiply $\frac{{(y - 3)}^{2}}{4}$ by $\frac{9}{9}$ .

$\frac{{(x - 1)}^{2} \cdot 4}{36} - \frac{{(y - 3)}^{2} \cdot 9}{4 \cdot 9} = 1$

Step 1.3.4

Multiply $4$ by $9$ .

$\frac{{(x - 1)}^{2} \cdot 4}{36} - \frac{{(y - 3)}^{2} \cdot 9}{36} = 1$

Step 1.4

Combine the numerators over the common denominator.

$\frac{{(x - 1)}^{2} \cdot 4 - {(y - 3)}^{2} \cdot 9}{36} = 1$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Rewrite ${(x - 1)}^{2} \cdot 4$ as ${((x - 1) \cdot 2)}^{2}$ .

$\frac{{((x - 1) \cdot 2)}^{2} - {(y - 3)}^{2} \cdot 9}{36} = 1$

Step 1.5.2

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

$\frac{{((x - 1) \cdot 2)}^{2} - {((y - 3) \cdot 3)}^{2}}{36} = 1$

Step 1.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = (x - 1) \cdot 2$ and $b = (y - 3) \cdot 3$ .

$\frac{((x - 1) \cdot 2 + (y - 3) \cdot 3) ((x - 1) \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4

Simplify.

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Step 1.5.4.1

Apply the distributive property.

$\frac{(x \cdot 2 - 1 \cdot 2 + (y - 3) \cdot 3) ((x - 1) \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.2

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

$\frac{(2 \cdot x - 1 \cdot 2 + (y - 3) \cdot 3) ((x - 1) \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.3

Multiply $- 1$ by $2$ .

$\frac{(2 \cdot x - 2 + (y - 3) \cdot 3) ((x - 1) \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.4

Apply the distributive property.

$\frac{(2 x - 2 + y \cdot 3 - 3 \cdot 3) ((x - 1) \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.5

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

$\frac{(2 x - 2 + 3 \cdot y - 3 \cdot 3) ((x - 1) \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.6

Multiply $- 3$ by $3$ .

$\frac{(2 x - 2 + 3 \cdot y - 9) ((x - 1) \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.7

Subtract $9$ from $- 2$ .

$\frac{(2 x + 3 y - 11) ((x - 1) \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.8

Apply the distributive property.

$\frac{(2 x + 3 y - 11) (x \cdot 2 - 1 \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.9

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

$\frac{(2 x + 3 y - 11) (2 \cdot x - 1 \cdot 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.10

Multiply $- 1$ by $2$ .

$\frac{(2 x + 3 y - 11) (2 \cdot x - 2 - ((y - 3) \cdot 3))}{36} = 1$

Step 1.5.4.11

Apply the distributive property.

$\frac{(2 x + 3 y - 11) (2 x - 2 - (y \cdot 3 - 3 \cdot 3))}{36} = 1$

Step 1.5.4.12

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

$\frac{(2 x + 3 y - 11) (2 x - 2 - (3 \cdot y - 3 \cdot 3))}{36} = 1$

Step 1.5.4.13

Multiply $- 3$ by $3$ .

$\frac{(2 x + 3 y - 11) (2 x - 2 - (3 \cdot y - 9))}{36} = 1$

Step 1.5.4.14

Apply the distributive property.

$\frac{(2 x + 3 y - 11) (2 x - 2 - (3 y) + 9)}{36} = 1$

Step 1.5.4.15

Multiply $3$ by $- 1$ .

$\frac{(2 x + 3 y - 11) (2 x - 2 - 3 y + 9)}{36} = 1$

Step 1.5.4.16

Multiply $- 1$ by $- 9$ .

$\frac{(2 x + 3 y - 11) (2 x - 2 - 3 y + 9)}{36} = 1$

Step 1.5.4.17

Add $- 2$ and $9$ .

$\frac{(2 x + 3 y - 11) (2 x - 3 y + 7)}{36} = 1$

Step 2

Multiply both sides by $36$ .

$\frac{(2 x + 3 y - 11) (2 x - 3 y + 7)}{36} \cdot 36 = 1 \cdot 36$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{(2 x + 3 y - 11) (2 x - 3 y + 7)}{36} \cdot 36$ .

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Step 3.1.1.1

Cancel the common factor of $36$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{(2 x + 3 y - 11) (2 x - 3 y + 7)}{36} \cdot 36 = 1 \cdot 36$

Step 3.1.1.1.2

Rewrite the expression.

$(2 x + 3 y - 11) (2 x - 3 y + 7) = 1 \cdot 36$

Step 3.1.1.2

Expand $(2 x + 3 y - 11) (2 x - 3 y + 7)$ by multiplying each term in the first expression by each term in the second expression.

$2 x (2 x) + 2 x (- 3 y) + 2 x \cdot 7 + 3 y (2 x) + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3

Simplify terms.

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Step 3.1.1.3.1

Simplify each term.

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Step 3.1.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x (- 3 y) + 2 x \cdot 7 + 3 y (2 x) + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.2

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

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Step 3.1.1.3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x (- 3 y) + 2 x \cdot 7 + 3 y (2 x) + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x (- 3 y) + 2 x \cdot 7 + 3 y (2 x) + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x (- 3 y) + 2 x \cdot 7 + 3 y (2 x) + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 x^{2} + 2 \cdot - 3 x y + 2 x \cdot 7 + 3 y (2 x) + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.5

Multiply $2$ by $- 3$ .

$4 x^{2} - 6 x y + 2 x \cdot 7 + 3 y (2 x) + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.6

Multiply $7$ by $2$ .

$4 x^{2} - 6 x y + 14 x + 3 y (2 x) + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.7

Rewrite using the commutative property of multiplication.

$4 x^{2} - 6 x y + 14 x + 3 \cdot 2 y x + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.8

Multiply $3$ by $2$ .

$4 x^{2} - 6 x y + 14 x + 6 y x + 3 y (- 3 y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.9

Rewrite using the commutative property of multiplication.

$4 x^{2} - 6 x y + 14 x + 6 y x + 3 \cdot - 3 y \cdot y + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.10

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.1.3.1.10.1

Move $y$ .

$4 x^{2} - 6 x y + 14 x + 6 y x + 3 \cdot - 3 (y \cdot y) + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.10.2

Multiply $y$ by $y$ .

$4 x^{2} - 6 x y + 14 x + 6 y x + 3 \cdot - 3 y^{2} + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.11

Multiply $3$ by $- 3$ .

$4 x^{2} - 6 x y + 14 x + 6 y x - 9 y^{2} + 3 y \cdot 7 - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.12

Multiply $7$ by $3$ .

$4 x^{2} - 6 x y + 14 x + 6 y x - 9 y^{2} + 21 y - 11 (2 x) - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.13

Multiply $2$ by $- 11$ .

$4 x^{2} - 6 x y + 14 x + 6 y x - 9 y^{2} + 21 y - 22 x - 11 (- 3 y) - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.14

Multiply $- 3$ by $- 11$ .

$4 x^{2} - 6 x y + 14 x + 6 y x - 9 y^{2} + 21 y - 22 x + 33 y - 11 \cdot 7 = 1 \cdot 36$

Step 3.1.1.3.1.15

Multiply $- 11$ by $7$ .

$4 x^{2} - 6 x y + 14 x + 6 y x - 9 y^{2} + 21 y - 22 x + 33 y - 77 = 1 \cdot 36$

Step 3.1.1.3.2

Simplify by adding terms.

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Step 3.1.1.3.2.1

Combine the opposite terms in $4 x^{2} - 6 x y + 14 x + 6 y x - 9 y^{2} + 21 y - 22 x + 33 y - 77$ .

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Step 3.1.1.3.2.1.1

Reorder the factors in the terms $- 6 x y$ and $6 y x$ .

$4 x^{2} - 6 x y + 14 x + 6 x y - 9 y^{2} + 21 y - 22 x + 33 y - 77 = 1 \cdot 36$

Step 3.1.1.3.2.1.2

Add $- 6 x y$ and $6 x y$ .

$4 x^{2} + 14 x + 0 - 9 y^{2} + 21 y - 22 x + 33 y - 77 = 1 \cdot 36$

Step 3.1.1.3.2.1.3

Add $4 x^{2} + 14 x$ and $0$ .

$4 x^{2} + 14 x - 9 y^{2} + 21 y - 22 x + 33 y - 77 = 1 \cdot 36$

Step 3.1.1.3.2.2

Subtract $22 x$ from $14 x$ .

$4 x^{2} - 8 x - 9 y^{2} + 21 y + 33 y - 77 = 1 \cdot 36$

Step 3.1.1.3.2.3

Add $21 y$ and $33 y$ .

$4 x^{2} - 8 x - 9 y^{2} + 54 y - 77 = 1 \cdot 36$

Step 3.1.1.3.2.4

Move $- 8 x$ .

$4 x^{2} - 9 y^{2} - 8 x + 54 y - 77 = 1 \cdot 36$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $36$ by $1$ .

$4 x^{2} - 9 y^{2} - 8 x + 54 y - 77 = 36$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract $36$ from both sides of the equation.

$4 x^{2} - 9 y^{2} - 8 x + 54 y - 77 - 36 = 0$

Step 4.2

Subtract $36$ from $- 77$ .

$4 x^{2} - 9 y^{2} - 8 x + 54 y - 113 = 0$