Trigonometry Examples

$9 y^{2} + 41 = 4 x^{2} + 54 y + 8 x$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Subtract $4 x^{2}$ from both sides of the equation.

$9 y^{2} + 41 - 4 x^{2} = 54 y + 8 x$

Step 1.2

Subtract $54 y$ from both sides of the equation.

$9 y^{2} + 41 - 4 x^{2} - 54 y = 8 x$

Step 1.3

Subtract $8 x$ from both sides of the equation.

$9 y^{2} + 41 - 4 x^{2} - 54 y - 8 x = 0$

Step 1.4

Move $41$ .

$9 y^{2} - 4 x^{2} - 54 y - 8 x + 41 = 0$

Step 1.5

Move $- 54 y$ .

$9 y^{2} - 4 x^{2} - 8 x - 54 y + 41 = 0$

Step 1.6

Reorder $9 y^{2}$ and $- 4 x^{2}$ .

$- 4 x^{2} + 9 y^{2} - 8 x - 54 y + 41 = 0$

Step 2

Subtract $41$ from both sides of the equation.

$- 4 x^{2} + 9 y^{2} - 8 x - 54 y = - 41$

Step 3

Complete the square for $- 4 x^{2} - 8 x$ .

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Step 3.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 4$

$b = - 8$

$c = 0$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 3.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 3.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 8}{2 \cdot - 4}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $- 8$ and $2$ .

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Step 3.3.2.1.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot - 4}$

Step 3.3.2.1.2

Cancel the common factors.

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Step 3.3.2.1.2.1

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot - 4}{2 (- 4)}$

Step 3.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot - 4}$

Step 3.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 4}{- 4}$

Step 3.3.2.2

Cancel the common factor of $- 4$ .

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Step 3.3.2.2.1

Cancel the common factor.

$d = \frac{- 4}{- 4}$

Step 3.3.2.2.2

Rewrite the expression.

$d = 1$

Step 3.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot - 4}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Raise $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot - 4}$

Step 3.4.2.1.2

Multiply $4$ by $- 4$ .

$e = 0 - \frac{64}{- 16}$

Step 3.4.2.1.3

Divide $64$ by $- 16$ .

$e = 0 - - 4$

Step 3.4.2.1.4

Multiply $- 1$ by $- 4$ .

$e = 0 + 4$

Step 3.4.2.2

Add $0$ and $4$ .

$e = 4$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x + 1)}^{2} + 4$ .

$- 4 {(x + 1)}^{2} + 4$

Step 4

Substitute $- 4 {(x + 1)}^{2} + 4$ for $- 4 x^{2} - 8 x$ in the equation $- 4 x^{2} + 9 y^{2} - 8 x - 54 y = - 41$ .

$- 4 {(x + 1)}^{2} + 4 + 9 y^{2} - 54 y = - 41$

Step 5

Move $4$ to the right side of the equation by adding $4$ to both sides.

$- 4 {(x + 1)}^{2} + 9 y^{2} - 54 y = - 41 - 4$

Step 6

Complete the square for $9 y^{2} - 54 y$ .

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Step 6.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 9$

$b = - 54$

$c = 0$

Step 6.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 6.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 6.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 54}{2 \cdot 9}$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Cancel the common factor of $- 54$ and $2$ .

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Step 6.3.2.1.1

Factor $2$ out of $- 54$ .

$d = \frac{2 \cdot - 27}{2 \cdot 9}$

Step 6.3.2.1.2

Cancel the common factors.

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Step 6.3.2.1.2.1

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 27}{2 (9)}$

Step 6.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 27}{2 \cdot 9}$

Step 6.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 27}{9}$

Step 6.3.2.2

Cancel the common factor of $- 27$ and $9$ .

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Step 6.3.2.2.1

Factor $9$ out of $- 27$ .

$d = \frac{9 \cdot - 3}{9}$

Step 6.3.2.2.2

Cancel the common factors.

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Step 6.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 3}{9 (1)}$

Step 6.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 3}{9 \cdot 1}$

Step 6.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 3}{1}$

Step 6.3.2.2.2.4

Divide $- 3$ by $1$ .

$d = - 3$

Step 6.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 6.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 54)}^{2}}{4 \cdot 9}$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Raise $- 54$ to the power of $2$ .

$e = 0 - \frac{2916}{4 \cdot 9}$

Step 6.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{2916}{36}$

Step 6.4.2.1.3

Divide $2916$ by $36$ .

$e = 0 - 1 \cdot 81$

Step 6.4.2.1.4

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Step 6.4.2.2

Subtract $81$ from $0$ .

$e = - 81$

Step 6.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(y - 3)}^{2} - 81$ .

$9 {(y - 3)}^{2} - 81$

Step 7

Substitute $9 {(y - 3)}^{2} - 81$ for $9 y^{2} - 54 y$ in the equation $- 4 x^{2} + 9 y^{2} - 8 x - 54 y = - 41$ .

$- 4 {(x + 1)}^{2} + 9 {(y - 3)}^{2} - 81 = - 41 - 4$

Step 8

Move $- 81$ to the right side of the equation by adding $81$ to both sides.

$- 4 {(x + 1)}^{2} + 9 {(y - 3)}^{2} = - 41 - 4 + 81$

Step 9

Simplify $- 41 - 4 + 81$ .

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Step 9.1

Subtract $4$ from $- 41$ .

$- 4 {(x + 1)}^{2} + 9 {(y - 3)}^{2} = - 45 + 81$

Step 9.2

Add $- 45$ and $81$ .

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

Step 10

Divide each term by $36$ to make the right side equal to one.

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

Step 11

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

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