Precalculus Examples

$\frac{{(x + 5)}^{2}}{5} + \frac{{(y + 9)}^{2}}{9}$

Step 1

Subtract $\frac{{(x + 5)}^{2}}{5}$ from both sides of the equation.

$y - \frac{{(x + 5)}^{2}}{5} = \frac{{(y + 9)}^{2}}{9}$

Step 2

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

$y - \frac{{(x + 5)}^{2}}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 3

To write $y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y \cdot \frac{5}{5} - \frac{{(x + 5)}^{2}}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 4

Combine $y$ and $\frac{5}{5}$ .

$\frac{y \cdot 5}{5} - \frac{{(x + 5)}^{2}}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{y \cdot 5 - {(x + 5)}^{2}}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6

Simplify each term.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Move $5$ to the left of $y$ .

$\frac{5 \cdot y - {(x + 5)}^{2}}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.2

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$\frac{5 y - ((x + 5) (x + 5))}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.3

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 6.1.3.1

Apply the distributive property.

$\frac{5 y - (x (x + 5) + 5 (x + 5))}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.3.2

Apply the distributive property.

$\frac{5 y - (x \cdot x + x \cdot 5 + 5 (x + 5))}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.3.3

Apply the distributive property.

$\frac{5 y - (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.4

Simplify and combine like terms.

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Step 6.1.4.1

Simplify each term.

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Step 6.1.4.1.1

Multiply $x$ by $x$ .

$\frac{5 y - (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.4.1.2

Move $5$ to the left of $x$ .

$\frac{5 y - (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.4.1.3

Multiply $5$ by $5$ .

$\frac{5 y - (x^{2} + 5 x + 5 x + 25)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.4.2

Add $5 x$ and $5 x$ .

$\frac{5 y - (x^{2} + 10 x + 25)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.5

Apply the distributive property.

$\frac{5 y - x^{2} - (10 x) - 1 \cdot 25}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.6

Simplify.

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Step 6.1.6.1

Multiply $10$ by $- 1$ .

$\frac{5 y - x^{2} - 10 x - 1 \cdot 25}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.6.2

Multiply $- 1$ by $25$ .

$\frac{5 y - x^{2} - 10 x - 25}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7

Rewrite $5 y - x^{2} - 10 x - 25$ in a factored form.

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Step 6.1.7.1

Regroup terms.

$\frac{- 25 + 5 y - x^{2} - 10 x}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.2

Factor $- 1$ out of $5 y - x^{2} - 10 x$ .

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Step 6.1.7.2.1

Reorder $5 y$ and $- x^{2}$ .

$\frac{- 25 - x^{2} + 5 y - 10 x}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.2.2

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

$\frac{- 25 - (x^{2}) + 5 y - 10 x}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.2.3

Factor $- 1$ out of $5 y$ .

$\frac{- 25 - (x^{2}) - (- 5 y) - 10 x}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.2.4

Factor $- 1$ out of $- 10 x$ .

$\frac{- 25 - (x^{2}) - (- 5 y) - (10 x)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.2.5

Factor $- 1$ out of $- (x^{2}) - (- 5 y)$ .

$\frac{- 25 - (x^{2} - 5 y) - (10 x)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.2.6

Factor $- 1$ out of $- (x^{2} - 5 y) - (10 x)$ .

$\frac{- 25 - (x^{2} - 5 y + 10 x)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.3

Factor $- 1$ out of $- 25 - (x^{2} - 5 y + 10 x)$ .

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Step 6.1.7.3.1

Reorder $- 25$ and $- (x^{2} - 5 y + 10 x)$ .

$\frac{- (x^{2} - 5 y + 10 x) - 25}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.3.2

Rewrite $- 25$ as $- 1 (25)$ .

$\frac{- (x^{2} - 5 y + 10 x) - 1 (25)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.1.7.3.3

Factor $- 1$ out of $- (x^{2} - 5 y + 10 x) - 1 (25)$ .

$\frac{- (x^{2} - 5 y + 10 x + 25)}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 6.2

Move the negative in front of the fraction.

$- \frac{x^{2} - 5 y + 10 x + 25}{5} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 7

To write $- \frac{x^{2} - 5 y + 10 x + 25}{5}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{x^{2} - 5 y + 10 x + 25}{5} \cdot \frac{9}{9} - \frac{{(y + 9)}^{2}}{9} = 0$

Step 8

To write $- \frac{{(y + 9)}^{2}}{9}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{x^{2} - 5 y + 10 x + 25}{5} \cdot \frac{9}{9} - \frac{{(y + 9)}^{2}}{9} \cdot \frac{5}{5} = 0$

Step 9

Write each expression with a common denominator of $45$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{x^{2} - 5 y + 10 x + 25}{5}$ by $\frac{9}{9}$ .

$- \frac{(x^{2} - 5 y + 10 x + 25) \cdot 9}{5 \cdot 9} - \frac{{(y + 9)}^{2}}{9} \cdot \frac{5}{5} = 0$

Step 9.2

Multiply $5$ by $9$ .

$- \frac{(x^{2} - 5 y + 10 x + 25) \cdot 9}{45} - \frac{{(y + 9)}^{2}}{9} \cdot \frac{5}{5} = 0$

Step 9.3

Multiply $\frac{{(y + 9)}^{2}}{9}$ by $\frac{5}{5}$ .

$- \frac{(x^{2} - 5 y + 10 x + 25) \cdot 9}{45} - \frac{{(y + 9)}^{2} \cdot 5}{9 \cdot 5} = 0$

Step 9.4

Multiply $9$ by $5$ .

$- \frac{(x^{2} - 5 y + 10 x + 25) \cdot 9}{45} - \frac{{(y + 9)}^{2} \cdot 5}{45} = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{- (x^{2} - 5 y + 10 x + 25) \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

$\frac{(- x^{2} - (- 5 y) - (10 x) - 1 \cdot 25) \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.2

Simplify.

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Step 11.2.1

Multiply $- 5$ by $- 1$ .

$\frac{(- x^{2} + 5 y - (10 x) - 1 \cdot 25) \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.2.2

Multiply $10$ by $- 1$ .

$\frac{(- x^{2} + 5 y - 10 x - 1 \cdot 25) \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.2.3

Multiply $- 1$ by $25$ .

$\frac{(- x^{2} + 5 y - 10 x - 25) \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.3

Apply the distributive property.

$\frac{- x^{2} \cdot 9 + 5 y \cdot 9 - 10 x \cdot 9 - 25 \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.4

Simplify.

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Step 11.4.1

Multiply $9$ by $- 1$ .

$\frac{- 9 x^{2} + 5 y \cdot 9 - 10 x \cdot 9 - 25 \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.4.2

Multiply $9$ by $5$ .

$\frac{- 9 x^{2} + 45 y - 10 x \cdot 9 - 25 \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.4.3

Multiply $9$ by $- 10$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 25 \cdot 9 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.4.4

Multiply $- 25$ by $9$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - {(y + 9)}^{2} \cdot 5}{45} = 0$

Step 11.5

Rewrite ${(y + 9)}^{2}$ as $(y + 9) (y + 9)$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - ((y + 9) (y + 9)) \cdot 5}{45} = 0$

Step 11.6

Expand $(y + 9) (y + 9)$ using the FOIL Method.

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Step 11.6.1

Apply the distributive property.

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - (y (y + 9) + 9 (y + 9)) \cdot 5}{45} = 0$

Step 11.6.2

Apply the distributive property.

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - (y \cdot y + y \cdot 9 + 9 (y + 9)) \cdot 5}{45} = 0$

Step 11.6.3

Apply the distributive property.

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - (y \cdot y + y \cdot 9 + 9 y + 9 \cdot 9) \cdot 5}{45} = 0$

Step 11.7

Simplify and combine like terms.

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Step 11.7.1

Simplify each term.

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Step 11.7.1.1

Multiply $y$ by $y$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - (y^{2} + y \cdot 9 + 9 y + 9 \cdot 9) \cdot 5}{45} = 0$

Step 11.7.1.2

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

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - (y^{2} + 9 \cdot y + 9 y + 9 \cdot 9) \cdot 5}{45} = 0$

Step 11.7.1.3

Multiply $9$ by $9$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - (y^{2} + 9 y + 9 y + 81) \cdot 5}{45} = 0$

Step 11.7.2

Add $9 y$ and $9 y$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - (y^{2} + 18 y + 81) \cdot 5}{45} = 0$

Step 11.8

Apply the distributive property.

$\frac{- 9 x^{2} + 45 y - 90 x - 225 + (- y^{2} - (18 y) - 1 \cdot 81) \cdot 5}{45} = 0$

Step 11.9

Simplify.

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Step 11.9.1

Multiply $18$ by $- 1$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 + (- y^{2} - 18 y - 1 \cdot 81) \cdot 5}{45} = 0$

Step 11.9.2

Multiply $- 1$ by $81$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 + (- y^{2} - 18 y - 81) \cdot 5}{45} = 0$

Step 11.10

Apply the distributive property.

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - y^{2} \cdot 5 - 18 y \cdot 5 - 81 \cdot 5}{45} = 0$

Step 11.11

Simplify.

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Step 11.11.1

Multiply $5$ by $- 1$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - 5 y^{2} - 18 y \cdot 5 - 81 \cdot 5}{45} = 0$

Step 11.11.2

Multiply $5$ by $- 18$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - 5 y^{2} - 90 y - 81 \cdot 5}{45} = 0$

Step 11.11.3

Multiply $- 81$ by $5$ .

$\frac{- 9 x^{2} + 45 y - 90 x - 225 - 5 y^{2} - 90 y - 405}{45} = 0$

Step 11.12

Subtract $90 y$ from $45 y$ .

$\frac{- 9 x^{2} - 90 x - 225 - 5 y^{2} - 45 y - 405}{45} = 0$

Step 11.13

Subtract $405$ from $- 225$ .

$\frac{- 9 x^{2} - 90 x - 5 y^{2} - 45 y - 630}{45} = 0$

Step 12

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

$\frac{- (9 x^{2}) - 90 x - 5 y^{2} - 45 y - 630}{45} = 0$

Step 13

Factor $- 1$ out of $- 90 x$ .

$\frac{- (9 x^{2}) - (90 x) - 5 y^{2} - 45 y - 630}{45} = 0$

Step 14

Factor $- 1$ out of $- (9 x^{2}) - (90 x)$ .

$\frac{- (9 x^{2} + 90 x) - 5 y^{2} - 45 y - 630}{45} = 0$

Step 15

Factor $- 1$ out of $- 5 y^{2}$ .

$\frac{- (9 x^{2} + 90 x) - (5 y^{2}) - 45 y - 630}{45} = 0$

Step 16

Factor $- 1$ out of $- (9 x^{2} + 90 x) - (5 y^{2})$ .

$\frac{- (9 x^{2} + 90 x + 5 y^{2}) - 45 y - 630}{45} = 0$

Step 17

Factor $- 1$ out of $- 45 y$ .

$\frac{- (9 x^{2} + 90 x + 5 y^{2}) - (45 y) - 630}{45} = 0$

Step 18

Factor $- 1$ out of $- (9 x^{2} + 90 x + 5 y^{2}) - (45 y)$ .

$\frac{- (9 x^{2} + 90 x + 5 y^{2} + 45 y) - 630}{45} = 0$

Step 19

Rewrite $- 630$ as $- 1 (630)$ .

$\frac{- (9 x^{2} + 90 x + 5 y^{2} + 45 y) - 1 (630)}{45} = 0$

Step 20

Factor $- 1$ out of $- (9 x^{2} + 90 x + 5 y^{2} + 45 y) - 1 (630)$ .

$\frac{- (9 x^{2} + 90 x + 5 y^{2} + 45 y + 630)}{45} = 0$

Step 21

Rewrite $- (9 x^{2} + 90 x + 5 y^{2} + 45 y + 630)$ as $- 1 (9 x^{2} + 90 x + 5 y^{2} + 45 y + 630)$ .

$\frac{- 1 (9 x^{2} + 90 x + 5 y^{2} + 45 y + 630)}{45} = 0$

Step 22

Move the negative in front of the fraction.

$- \frac{9 x^{2} + 90 x + 5 y^{2} + 45 y + 630}{45} = 0$