Algebra Examples

$9 y^{2} - x^{2} + 54 y + 4 x + 68 = 0$

Step 1

Subtract $68$ from both sides of the equation.

$9 y^{2} - x^{2} + 54 y + 4 x = - 68$

Step 2

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

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

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

Step 3

Substitute $9 {(y + 3)}^{2} - 81$ for $9 y^{2} + 54 y$ in the equation $9 y^{2} - x^{2} + 54 y + 4 x = - 68$ .

$9 {(y + 3)}^{2} - 81 - x^{2} + 4 x = - 68$

Step 4

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

$9 {(y + 3)}^{2} - x^{2} + 4 x = - 68 + 81$

Step 5

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

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Step 5.1

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

$a = - 1$

$b = 4$

$c = 0$

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

$d = \frac{4}{2 \cdot - 1}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $4$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2 \cdot - 1}$

Step 5.3.2.1.2

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Step 5.3.2.2

Multiply $- 1$ by $2$ .

$d = - 2$

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

$e = 0 - \frac{4^{2}}{4 \cdot - 1}$

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

Cancel the common factor of $4^{2}$ and $4$ .

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Step 5.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot - 1}$

Step 5.4.2.1.1.2

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$e = 0 - (- 1 \cdot 4)$

Step 5.4.2.1.2

Multiply $- (- 1 \cdot 4)$ .

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Step 5.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - - 4$

Step 5.4.2.1.2.2

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

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

Step 6

Substitute $- {(x - 2)}^{2} + 4$ for $- x^{2} + 4 x$ in the equation $9 y^{2} - x^{2} + 54 y + 4 x = - 68$ .

$9 {(y + 3)}^{2} - {(x - 2)}^{2} + 4 = - 68 + 81$

Step 7

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

$9 {(y + 3)}^{2} - {(x - 2)}^{2} = - 68 + 81 - 4$

Step 8

Simplify $- 68 + 81 - 4$ .

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Step 8.1

Add $- 68$ and $81$ .

$9 {(y + 3)}^{2} - {(x - 2)}^{2} = 13 - 4$

Step 8.2

Subtract $4$ from $13$ .

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

Step 9

Divide each term by $9$ to make the right side equal to one.

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

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 $1$ .

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

Step 11

This is the standard form of a hyperbola. Use this form to determine the eccentricity.

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

Step 12

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = 1$

$b = 3$

$k = - 3$

$h = 2$

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.

$\frac{\sqrt{{(1)}^{2} + {(3)}^{2}}}{1}$

Step 15

Simplify.

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Step 15.1

Divide $\sqrt{{(1)}^{2} + {(3)}^{2}}$ by $1$ .

$\sqrt{{(1)}^{2} + {(3)}^{2}}$

Step 15.2

One to any power is one.

$\sqrt{1 + {(3)}^{2}}$

Step 15.3

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

$\sqrt{1 + 9}$

Step 15.4

Add $1$ and $9$ .

$\sqrt{10}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\sqrt{10}$

Decimal Form:

$3.16227766 \dots$

Step 17