Precalculus Examples

$6 x^{2} - 12 x + 54 y^{2} + 108 y - 426 = 0$

Step 1

Add $426$ to both sides of the equation.

$6 x^{2} - 12 x + 54 y^{2} + 108 y = 426$

Step 2

Complete the square for $6 x^{2} - 12 x$ .

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Step 2.1

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

$a = 6$

$b = - 12$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 12}{2 \cdot 6}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 12$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot 6}$

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 $2 \cdot 6$ .

$d = \frac{2 \cdot - 6}{2 (6)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot 6}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 6}{6}$

Step 2.3.2.2

Cancel the common factor of $- 6$ and $6$ .

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Step 2.3.2.2.1

Factor $6$ out of $- 6$ .

$d = \frac{6 \cdot - 1}{6}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $6$ out of $6$ .

$d = \frac{6 \cdot - 1}{6 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{6 \cdot - 1}{6 \cdot 1}$

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 12)}^{2}}{4 \cdot 6}$

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 $- 12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot 6}$

Step 2.4.2.1.2

Multiply $4$ by $6$ .

$e = 0 - \frac{144}{24}$

Step 2.4.2.1.3

Divide $144$ by $24$ .

$e = 0 - 1 \cdot 6$

Step 2.4.2.1.4

Multiply $- 1$ by $6$ .

$e = 0 - 6$

Step 2.4.2.2

Subtract $6$ from $0$ .

$e = - 6$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $6 {(x - 1)}^{2} - 6$ .

$6 {(x - 1)}^{2} - 6$

Step 3

Substitute $6 {(x - 1)}^{2} - 6$ for $6 x^{2} - 12 x$ in the equation $6 x^{2} - 12 x + 54 y^{2} + 108 y = 426$ .

$6 {(x - 1)}^{2} - 6 + 54 y^{2} + 108 y = 426$

Step 4

Move $- 6$ to the right side of the equation by adding $6$ to both sides.

$6 {(x - 1)}^{2} + 54 y^{2} + 108 y = 426 + 6$

Step 5

Complete the square for $54 y^{2} + 108 y$ .

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Step 5.1

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

$a = 54$

$b = 108$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{108}{2 \cdot 54}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $108$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $108$ .

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

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 $2 \cdot 54$ .

$d = \frac{2 \cdot 54}{2 (54)}$

Step 5.3.2.1.2.2

Cancel the common factor.

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

Step 5.3.2.1.2.3

Rewrite the expression.

$d = \frac{54}{54}$

Step 5.3.2.2

Cancel the common factor of $54$ .

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Step 5.3.2.2.1

Cancel the common factor.

$d = \frac{54}{54}$

Step 5.3.2.2.2

Rewrite the expression.

$d = 1$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{108^{2}}{4 \cdot 54}$

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 $108$ to the power of $2$ .

$e = 0 - \frac{11664}{4 \cdot 54}$

Step 5.4.2.1.2

Multiply $4$ by $54$ .

$e = 0 - \frac{11664}{216}$

Step 5.4.2.1.3

Divide $11664$ by $216$ .

$e = 0 - 1 \cdot 54$

Step 5.4.2.1.4

Multiply $- 1$ by $54$ .

$e = 0 - 54$

Step 5.4.2.2

Subtract $54$ from $0$ .

$e = - 54$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $54 {(y + 1)}^{2} - 54$ .

$54 {(y + 1)}^{2} - 54$

Step 6

Substitute $54 {(y + 1)}^{2} - 54$ for $54 y^{2} + 108 y$ in the equation $6 x^{2} - 12 x + 54 y^{2} + 108 y = 426$ .

$6 {(x - 1)}^{2} + 54 {(y + 1)}^{2} - 54 = 426 + 6$

Step 7

Move $- 54$ to the right side of the equation by adding $54$ to both sides.

$6 {(x - 1)}^{2} + 54 {(y + 1)}^{2} = 426 + 6 + 54$

Step 8

Simplify $426 + 6 + 54$ .

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Step 8.1

Add $426$ and $6$ .

$6 {(x - 1)}^{2} + 54 {(y + 1)}^{2} = 432 + 54$

Step 8.2

Add $432$ and $54$ .

$6 {(x - 1)}^{2} + 54 {(y + 1)}^{2} = 486$

Step 9

Divide each term by $486$ to make the right side equal to one.

$\frac{6 {(x - 1)}^{2}}{486} + \frac{54 {(y + 1)}^{2}}{486} = \frac{486}{486}$

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 $1$ .

$\frac{{(x - 1)}^{2}}{81} + \frac{{(y + 1)}^{2}}{9} = 1$

Step 11

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 12

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = 9$

$b = 3$

$k = - 1$

$h = 1$

Step 13

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Step 14

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(9)}^{2} - {(3)}^{2}}}{9}$

Step 15

Simplify.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Raise $9$ to the power of $2$ .

$\frac{\sqrt{81 - 3^{2}}}{9}$

Step 15.1.2

Raise $3$ to the power of $2$ .

$\frac{\sqrt{81 - 1 \cdot 9}}{9}$

Step 15.1.3

Multiply $- 1$ by $9$ .

$\frac{\sqrt{81 - 9}}{9}$

Step 15.1.4

Subtract $9$ from $81$ .

$\frac{\sqrt{72}}{9}$

Step 15.1.5

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 15.1.5.1

Factor $36$ out of $72$ .

$\frac{\sqrt{36 (2)}}{9}$

Step 15.1.5.2

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2} \cdot 2}}{9}$

Step 15.1.6

Pull terms out from under the radical.

$\frac{6 \sqrt{2}}{9}$

Step 15.2

Cancel the common factor of $6$ and $9$ .

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Step 15.2.1

Factor $3$ out of $6 \sqrt{2}$ .

$\frac{3 (2 \sqrt{2})}{9}$

Step 15.2.2

Cancel the common factors.

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Step 15.2.2.1

Factor $3$ out of $9$ .

$\frac{3 (2 \sqrt{2})}{3 (3)}$

Step 15.2.2.2

Cancel the common factor.

$\frac{3 (2 \sqrt{2})}{3 \cdot 3}$

Step 15.2.2.3

Rewrite the expression.

$\frac{2 \sqrt{2}}{3}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{2}}{3}$

Decimal Form:

$0.94280904 \dots$

Step 17