Precalculus Examples

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

Step 1

Subtract $41$ from both sides of the equation.

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

Step 2

Complete the square for $9 x^{2} + 54 x$ .

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Step 2.1

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 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Factor $2$ out of $54$ .

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

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

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

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

Step 2.3.2.1.2.2

Cancel the common factor.

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

Step 2.3.2.1.2.3

Rewrite the expression.

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

Factor $9$ out of $27$ .

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

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $9$ out of $9$ .

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

Step 2.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide $3$ by $1$ .

$d = 3$

Step 2.4

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

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Step 2.4.1

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 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

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

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

Step 2.4.2.1.2

Multiply $4$ by $9$ .

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

Step 2.4.2.1.3

Divide $2916$ by $36$ .

$e = 0 - 1 \cdot 81$

Step 2.4.2.1.4

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Step 2.4.2.2

Subtract $81$ from $0$ .

$e = - 81$

Step 2.5

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

$9 {(x + 3)}^{2} - 81$

Step 3

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

$9 {(x + 3)}^{2} - 81 - 4 y^{2} + 8 y = - 41$

Step 4

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

$9 {(x + 3)}^{2} - 4 y^{2} + 8 y = - 41 + 81$

Step 5

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

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Step 5.1

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 5.2

Consider the vertex form of a parabola.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

Factor $2$ out of $8$ .

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

Step 5.3.2.1.2

Cancel the common factors.

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Step 5.3.2.1.2.1

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

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

Step 5.3.2.1.2.2

Cancel the common factor.

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

Step 5.3.2.1.2.3

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $4$ and $- 4$ .

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Step 5.3.2.2.1

Factor $4$ out of $4$ .

$d = \frac{4 (1)}{- 4}$

Step 5.3.2.2.2

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Step 5.3.2.3

Multiply $- 1$ by $1$ .

$d = - 1$

Step 5.4

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

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Step 5.4.1

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 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

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

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

Step 5.4.2.1.2

Multiply $4$ by $- 4$ .

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

Step 5.4.2.1.3

Divide $64$ by $- 16$ .

$e = 0 - - 4$

Step 5.4.2.1.4

Multiply $- 1$ by $- 4$ .

$e = 0 + 4$

Step 5.4.2.2

Add $0$ and $4$ .

$e = 4$

Step 5.5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify $- 41 + 81 - 4$ .

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Step 8.1

Add $- 41$ and $81$ .

$9 {(x + 3)}^{2} - 4 {(y - 1)}^{2} = 40 - 4$

Step 8.2

Subtract $4$ from $40$ .

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

Step 9

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

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

Step 10

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 $1$ .

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