Precalculus Examples

$\frac{{(x - 4)}^{2}}{16} - \frac{{(y + 3)}^{2}}{4} = 1$

Step 1

Simplify the left side $\frac{{(x - 4)}^{2}}{16} - \frac{{(y + 3)}^{2}}{4}$ .

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Step 1.1

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

$\frac{{(x - 4)}^{2}}{16} - \frac{{(y + 3)}^{2}}{4} \cdot \frac{4}{4} = 1$

Step 1.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

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

$\frac{{(x - 4)}^{2}}{16} - \frac{{(y + 3)}^{2} \cdot 4}{4 \cdot 4} = 1$

Step 1.2.2

Multiply $4$ by $4$ .

$\frac{{(x - 4)}^{2}}{16} - \frac{{(y + 3)}^{2} \cdot 4}{16} = 1$

Step 1.3

Combine the numerators over the common denominator.

$\frac{{(x - 4)}^{2} - {(y + 3)}^{2} \cdot 4}{16} = 1$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Rewrite ${(y + 3)}^{2} \cdot 4$ as ${((y + 3) \cdot 2)}^{2}$ .

$\frac{{(x - 4)}^{2} - {((y + 3) \cdot 2)}^{2}}{16} = 1$

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 $a = x - 4$ and $b = (y + 3) \cdot 2$ .

$\frac{(x - 4 + (y + 3) \cdot 2) (x - 4 - ((y + 3) \cdot 2))}{16} = 1$

Step 1.4.3

Simplify.

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Step 1.4.3.1

Apply the distributive property.

$\frac{(x - 4 + y \cdot 2 + 3 \cdot 2) (x - 4 - ((y + 3) \cdot 2))}{16} = 1$

Step 1.4.3.2

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

$\frac{(x - 4 + 2 \cdot y + 3 \cdot 2) (x - 4 - ((y + 3) \cdot 2))}{16} = 1$

Step 1.4.3.3

Multiply $3$ by $2$ .

$\frac{(x - 4 + 2 \cdot y + 6) (x - 4 - ((y + 3) \cdot 2))}{16} = 1$

Step 1.4.3.4

Add $- 4$ and $6$ .

$\frac{(x + 2 y + 2) (x - 4 - ((y + 3) \cdot 2))}{16} = 1$

Step 1.4.3.5

Apply the distributive property.

$\frac{(x + 2 y + 2) (x - 4 - (y \cdot 2 + 3 \cdot 2))}{16} = 1$

Step 1.4.3.6

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

$\frac{(x + 2 y + 2) (x - 4 - (2 \cdot y + 3 \cdot 2))}{16} = 1$

Step 1.4.3.7

Multiply $3$ by $2$ .

$\frac{(x + 2 y + 2) (x - 4 - (2 \cdot y + 6))}{16} = 1$

Step 1.4.3.8

Apply the distributive property.

$\frac{(x + 2 y + 2) (x - 4 - (2 y) - 1 \cdot 6)}{16} = 1$

Step 1.4.3.9

Multiply $2$ by $- 1$ .

$\frac{(x + 2 y + 2) (x - 4 - 2 y - 1 \cdot 6)}{16} = 1$

Step 1.4.3.10

Multiply $- 1$ by $6$ .

$\frac{(x + 2 y + 2) (x - 4 - 2 y - 6)}{16} = 1$

Step 1.4.3.11

Subtract $6$ from $- 4$ .

$\frac{(x + 2 y + 2) (x - 2 y - 10)}{16} = 1$

Step 2

Multiply both sides by $16$ .

$\frac{(x + 2 y + 2) (x - 2 y - 10)}{16} \cdot 16 = 1 \cdot 16$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{(x + 2 y + 2) (x - 2 y - 10)}{16} \cdot 16$ .

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Step 3.1.1.1

Cancel the common factor of $16$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{(x + 2 y + 2) (x - 2 y - 10)}{16} \cdot 16 = 1 \cdot 16$

Step 3.1.1.1.2

Rewrite the expression.

$(x + 2 y + 2) (x - 2 y - 10) = 1 \cdot 16$

Step 3.1.1.2

Expand $(x + 2 y + 2) (x - 2 y - 10)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x (- 2 y) + x \cdot - 10 + 2 y x + 2 y (- 2 y) + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3

Simplify terms.

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Step 3.1.1.3.1

Combine the opposite terms in $x \cdot x + x (- 2 y) + x \cdot - 10 + 2 y x + 2 y (- 2 y) + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10$ .

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Step 3.1.1.3.1.1

Reorder the factors in the terms $x (- 2 y)$ and $2 y x$ .

$x \cdot x - 2 x y + x \cdot - 10 + 2 x y + 2 y (- 2 y) + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.1.2

Add $- 2 x y$ and $2 x y$ .

$x \cdot x + x \cdot - 10 + 0 + 2 y (- 2 y) + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.1.3

Add $x \cdot x + x \cdot - 10$ and $0$ .

$x \cdot x + x \cdot - 10 + 2 y (- 2 y) + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2

Simplify each term.

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Step 3.1.1.3.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 10 + 2 y (- 2 y) + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2.2

Move $- 10$ to the left of $x$ .

$x^{2} - 10 \cdot x + 2 y (- 2 y) + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2.3

Rewrite using the commutative property of multiplication.

$x^{2} - 10 x + 2 \cdot - 2 y \cdot y + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2.4

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

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Step 3.1.1.3.2.4.1

Move $y$ .

$x^{2} - 10 x + 2 \cdot - 2 (y \cdot y) + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2.4.2

Multiply $y$ by $y$ .

$x^{2} - 10 x + 2 \cdot - 2 y^{2} + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2.5

Multiply $2$ by $- 2$ .

$x^{2} - 10 x - 4 y^{2} + 2 y \cdot - 10 + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2.6

Multiply $- 10$ by $2$ .

$x^{2} - 10 x - 4 y^{2} - 20 y + 2 x + 2 (- 2 y) + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2.7

Multiply $- 2$ by $2$ .

$x^{2} - 10 x - 4 y^{2} - 20 y + 2 x - 4 y + 2 \cdot - 10 = 1 \cdot 16$

Step 3.1.1.3.2.8

Multiply $2$ by $- 10$ .

$x^{2} - 10 x - 4 y^{2} - 20 y + 2 x - 4 y - 20 = 1 \cdot 16$

Step 3.1.1.3.3

Simplify by adding terms.

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Step 3.1.1.3.3.1

Add $- 10 x$ and $2 x$ .

$x^{2} - 8 x - 4 y^{2} - 20 y - 4 y - 20 = 1 \cdot 16$

Step 3.1.1.3.3.2

Subtract $4 y$ from $- 20 y$ .

$x^{2} - 8 x - 4 y^{2} - 24 y - 20 = 1 \cdot 16$

Step 3.1.1.3.3.3

Move $- 8 x$ .

$x^{2} - 4 y^{2} - 8 x - 24 y - 20 = 1 \cdot 16$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $16$ by $1$ .

$x^{2} - 4 y^{2} - 8 x - 24 y - 20 = 16$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract $16$ from both sides of the equation.

$x^{2} - 4 y^{2} - 8 x - 24 y - 20 - 16 = 0$

Step 4.2

Subtract $16$ from $- 20$ .

$x^{2} - 4 y^{2} - 8 x - 24 y - 36 = 0$