Trigonometry Examples

${(y - 2)}^{2} = 22 - \frac{22 {(x - 5)}^{2}}{25}$

Step 1

Add $\frac{22 {(x - 5)}^{2}}{25}$ to both sides of the equation.

${(y - 2)}^{2} + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 2

Simplify each term.

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Step 2.1

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

$(y - 2) (y - 2) + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 2.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$y (y - 2) - 2 (y - 2) + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 2.2.2

Apply the distributive property.

$y \cdot y + y \cdot - 2 - 2 (y - 2) + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 2.2.3

Apply the distributive property.

$y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2 + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $y$ by $y$ .

$y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2 + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 2.3.1.2

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

$y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2 + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 2.3.1.3

Multiply $- 2$ by $- 2$ .

$y^{2} - 2 y - 2 y + 4 + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 2.3.2

Subtract $2 y$ from $- 2 y$ .

$y^{2} - 4 y + 4 + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 3

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$- 4 y + 4 + y^{2} \cdot \frac{25}{25} + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 4

Combine $y^{2}$ and $\frac{25}{25}$ .

$- 4 y + 4 + \frac{y^{2} \cdot 25}{25} + \frac{22 {(x - 5)}^{2}}{25} = 22$

Step 5

Combine the numerators over the common denominator.

$- 4 y + 4 + \frac{y^{2} \cdot 25 + 22 {(x - 5)}^{2}}{25} = 22$

Step 6

Simplify the numerator.

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Step 6.1

Move $25$ to the left of $y^{2}$ .

$- 4 y + 4 + \frac{25 \cdot y^{2} + 22 {(x - 5)}^{2}}{25} = 22$

Step 6.2

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

$- 4 y + 4 + \frac{25 y^{2} + 22 ((x - 5) (x - 5))}{25} = 22$

Step 6.3

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

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Step 6.3.1

Apply the distributive property.

$- 4 y + 4 + \frac{25 y^{2} + 22 (x (x - 5) - 5 (x - 5))}{25} = 22$

Step 6.3.2

Apply the distributive property.

$- 4 y + 4 + \frac{25 y^{2} + 22 (x \cdot x + x \cdot - 5 - 5 (x - 5))}{25} = 22$

Step 6.3.3

Apply the distributive property.

$- 4 y + 4 + \frac{25 y^{2} + 22 (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5)}{25} = 22$

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $x$ by $x$ .

$- 4 y + 4 + \frac{25 y^{2} + 22 (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5)}{25} = 22$

Step 6.4.1.2

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

$- 4 y + 4 + \frac{25 y^{2} + 22 (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5)}{25} = 22$

Step 6.4.1.3

Multiply $- 5$ by $- 5$ .

$- 4 y + 4 + \frac{25 y^{2} + 22 (x^{2} - 5 x - 5 x + 25)}{25} = 22$

Step 6.4.2

Subtract $5 x$ from $- 5 x$ .

$- 4 y + 4 + \frac{25 y^{2} + 22 (x^{2} - 10 x + 25)}{25} = 22$

Step 6.5

Apply the distributive property.

$- 4 y + 4 + \frac{25 y^{2} + 22 x^{2} + 22 (- 10 x) + 22 \cdot 25}{25} = 22$

Step 6.6

Simplify.

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Step 6.6.1

Multiply $- 10$ by $22$ .

$- 4 y + 4 + \frac{25 y^{2} + 22 x^{2} - 220 x + 22 \cdot 25}{25} = 22$

Step 6.6.2

Multiply $22$ by $25$ .

$- 4 y + 4 + \frac{25 y^{2} + 22 x^{2} - 220 x + 550}{25} = 22$

Step 7

To write $- 4 y$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$- 4 y \cdot \frac{25}{25} + \frac{25 y^{2} + 22 x^{2} - 220 x + 550}{25} + 4 = 22$

Step 8

Combine $- 4 y$ and $\frac{25}{25}$ .

$\frac{- 4 y \cdot 25}{25} + \frac{25 y^{2} + 22 x^{2} - 220 x + 550}{25} + 4 = 22$

Step 9

Combine the numerators over the common denominator.

$\frac{- 4 y \cdot 25 + 25 y^{2} + 22 x^{2} - 220 x + 550}{25} + 4 = 22$

Step 10

Multiply $25$ by $- 4$ .

$\frac{- 100 y + 25 y^{2} + 22 x^{2} - 220 x + 550}{25} + 4 = 22$

Step 11

To write $4$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{- 100 y + 25 y^{2} + 22 x^{2} - 220 x + 550}{25} + 4 \cdot \frac{25}{25} = 22$

Step 12

Combine $4$ and $\frac{25}{25}$ .

$\frac{- 100 y + 25 y^{2} + 22 x^{2} - 220 x + 550}{25} + \frac{4 \cdot 25}{25} = 22$

Step 13

Combine the numerators over the common denominator.

$\frac{- 100 y + 25 y^{2} + 22 x^{2} - 220 x + 550 + 4 \cdot 25}{25} = 22$

Step 14

Simplify the numerator.

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Step 14.1

Multiply $4$ by $25$ .

$\frac{- 100 y + 25 y^{2} + 22 x^{2} - 220 x + 550 + 100}{25} = 22$

Step 14.2

Add $550$ and $100$ .

$\frac{- 100 y + 25 y^{2} + 22 x^{2} - 220 x + 650}{25} = 22$

Step 15

Factor $- 1$ out of $- 100 y$ .

$\frac{- (100 y) + 25 y^{2} + 22 x^{2} - 220 x + 650}{25} = 22$

Step 16

Factor $- 1$ out of $25 y^{2}$ .

$\frac{- (100 y) - (- 25 y^{2}) + 22 x^{2} - 220 x + 650}{25} = 22$

Step 17

Factor $- 1$ out of $- (100 y) - (- 25 y^{2})$ .

$\frac{- (100 y - 25 y^{2}) + 22 x^{2} - 220 x + 650}{25} = 22$

Step 18

Factor $- 1$ out of $22 x^{2}$ .

$\frac{- (100 y - 25 y^{2}) - (- 22 x^{2}) - 220 x + 650}{25} = 22$

Step 19

Factor $- 1$ out of $- (100 y - 25 y^{2}) - (- 22 x^{2})$ .

$\frac{- (100 y - 25 y^{2} - 22 x^{2}) - 220 x + 650}{25} = 22$

Step 20

Factor $- 1$ out of $- 220 x$ .

$\frac{- (100 y - 25 y^{2} - 22 x^{2}) - (220 x) + 650}{25} = 22$

Step 21

Factor $- 1$ out of $- (100 y - 25 y^{2} - 22 x^{2}) - (220 x)$ .

$\frac{- (100 y - 25 y^{2} - 22 x^{2} + 220 x) + 650}{25} = 22$

Step 22

Rewrite $650$ as $- 1 (- 650)$ .

$\frac{- (100 y - 25 y^{2} - 22 x^{2} + 220 x) - 1 (- 650)}{25} = 22$

Step 23

Factor $- 1$ out of $- (100 y - 25 y^{2} - 22 x^{2} + 220 x) - 1 (- 650)$ .

$\frac{- (100 y - 25 y^{2} - 22 x^{2} + 220 x - 650)}{25} = 22$

Step 24

Rewrite $- (100 y - 25 y^{2} - 22 x^{2} + 220 x - 650)$ as $- 1 (100 y - 25 y^{2} - 22 x^{2} + 220 x - 650)$ .

$\frac{- 1 (100 y - 25 y^{2} - 22 x^{2} + 220 x - 650)}{25} = 22$

Step 25

Move the negative in front of the fraction.

$- \frac{100 y - 25 y^{2} - 22 x^{2} + 220 x - 650}{25} = 22$

Step 26