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 ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Step 1.5.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.5.2.1

Apply the distributive property.

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

Step 1.5.2.2

Apply the distributive property.

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

Step 1.5.2.3

Apply the distributive property.

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

Step 1.5.3

Simplify and combine like terms.

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Step 1.5.3.1

Simplify each term.

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Step 1.5.3.1.1

Multiply $x$ by $x$ .

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

Step 1.5.3.1.2

Multiply $x$ by $1$ .

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

Step 1.5.3.1.3

Multiply $x$ by $1$ .

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

Step 1.5.3.1.4

Multiply $1$ by $1$ .

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

Step 1.5.3.2

Add $x$ and $x$ .

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

Step 1.5.4

Apply the distributive property.

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

Step 1.5.5

Simplify.

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Step 1.5.5.1

Move $4$ to the left of $x^{2}$ .

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

Step 1.5.5.2

Multiply $4$ by $2$ .

$\frac{4 \cdot x^{2} + 8 x + 1 \cdot 4 + {(y + 3)}^{2} \cdot 9}{36} = 1$

Step 1.5.5.3

Multiply $4$ by $1$ .

$\frac{4 \cdot x^{2} + 8 x + 4 + {(y + 3)}^{2} \cdot 9}{36} = 1$

Step 1.5.6

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

$\frac{4 x^{2} + 8 x + 4 + (y + 3) (y + 3) \cdot 9}{36} = 1$

Step 1.5.7

Expand $(y + 3) (y + 3)$ using the FOIL Method.

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Step 1.5.7.1

Apply the distributive property.

$\frac{4 x^{2} + 8 x + 4 + (y (y + 3) + 3 (y + 3)) \cdot 9}{36} = 1$

Step 1.5.7.2

Apply the distributive property.

$\frac{4 x^{2} + 8 x + 4 + (y \cdot y + y \cdot 3 + 3 (y + 3)) \cdot 9}{36} = 1$

Step 1.5.7.3

Apply the distributive property.

$\frac{4 x^{2} + 8 x + 4 + (y \cdot y + y \cdot 3 + 3 y + 3 \cdot 3) \cdot 9}{36} = 1$

Step 1.5.8

Simplify and combine like terms.

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Step 1.5.8.1

Simplify each term.

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Step 1.5.8.1.1

Multiply $y$ by $y$ .

$\frac{4 x^{2} + 8 x + 4 + (y^{2} + y \cdot 3 + 3 y + 3 \cdot 3) \cdot 9}{36} = 1$

Step 1.5.8.1.2

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

$\frac{4 x^{2} + 8 x + 4 + (y^{2} + 3 \cdot y + 3 y + 3 \cdot 3) \cdot 9}{36} = 1$

Step 1.5.8.1.3

Multiply $3$ by $3$ .

$\frac{4 x^{2} + 8 x + 4 + (y^{2} + 3 y + 3 y + 9) \cdot 9}{36} = 1$

Step 1.5.8.2

Add $3 y$ and $3 y$ .

$\frac{4 x^{2} + 8 x + 4 + (y^{2} + 6 y + 9) \cdot 9}{36} = 1$

Step 1.5.9

Apply the distributive property.

$\frac{4 x^{2} + 8 x + 4 + y^{2} \cdot 9 + 6 y \cdot 9 + 9 \cdot 9}{36} = 1$

Step 1.5.10

Simplify.

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Step 1.5.10.1

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

$\frac{4 x^{2} + 8 x + 4 + 9 \cdot y^{2} + 6 y \cdot 9 + 9 \cdot 9}{36} = 1$

Step 1.5.10.2

Multiply $9$ by $6$ .

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

Step 1.5.10.3

Multiply $9$ by $9$ .

$\frac{4 x^{2} + 8 x + 4 + 9 \cdot y^{2} + 54 y + 81}{36} = 1$

Step 1.5.11

Add $4$ and $81$ .

$\frac{4 x^{2} + 8 x + 9 y^{2} + 54 y + 85}{36} = 1$

Step 2

Multiply both sides by $36$ .

$\frac{4 x^{2} + 8 x + 9 y^{2} + 54 y + 85}{36} \cdot 36 = 1 \cdot 36$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{4 x^{2} + 8 x + 9 y^{2} + 54 y + 85}{36} \cdot 36$ .

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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.

$\frac{4 x^{2} + 8 x + 9 y^{2} + 54 y + 85}{36} \cdot 36 = 1 \cdot 36$

Step 3.1.1.1.2

Rewrite the expression.

$4 x^{2} + 8 x + 9 y^{2} + 54 y + 85 = 1 \cdot 36$

Step 3.1.1.2

Move $8 x$ .

$4 x^{2} + 9 y^{2} + 8 x + 54 y + 85 = 1 \cdot 36$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $36$ by $1$ .

$4 x^{2} + 9 y^{2} + 8 x + 54 y + 85 = 36$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract $36$ from both sides of the equation.

$4 x^{2} + 9 y^{2} + 8 x + 54 y + 85 - 36 = 0$

Step 4.2

Subtract $36$ from $85$ .

$4 x^{2} + 9 y^{2} + 8 x + 54 y + 49 = 0$

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