Algebra Examples

$9 x^{2} - 36 x - 4 y^{2} + 24 y - 36 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Add $36$ to both sides of the equation.

$9 x^{2} - 36 x - 4 y^{2} + 24 y = 36$

Step 1.2

Complete the square for $9 x^{2} - 36 x$ .

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Step 1.2.1

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

$a = 9$

$b = - 36$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{- 36}{2 \cdot 9}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 36$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 36$ .

$d = \frac{2 \cdot - 18}{2 \cdot 9}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

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

$d = \frac{2 \cdot - 18}{2 (9)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 18}{2 \cdot 9}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 18}{9}$

Step 1.2.3.2.2

Cancel the common factor of $- 18$ and $9$ .

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Step 1.2.3.2.2.1

Factor $9$ out of $- 18$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 2}{9 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 2}{9 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = 0 - \frac{{(- 36)}^{2}}{4 \cdot 9}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raise $- 36$ to the power of $2$ .

$e = 0 - \frac{1296}{4 \cdot 9}$

Step 1.2.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{1296}{36}$

Step 1.2.4.2.1.3

Divide $1296$ by $36$ .

$e = 0 - 1 \cdot 36$

Step 1.2.4.2.1.4

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Step 1.2.4.2.2

Subtract $36$ from $0$ .

$e = - 36$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 2)}^{2} - 36$ .

$9 {(x - 2)}^{2} - 36$

Step 1.3

Substitute $9 {(x - 2)}^{2} - 36$ for $9 x^{2} - 36 x$ in the equation $9 x^{2} - 36 x - 4 y^{2} + 24 y = 36$ .

$9 {(x - 2)}^{2} - 36 - 4 y^{2} + 24 y = 36$

Step 1.4

Move $- 36$ to the right side of the equation by adding $36$ to both sides.

$9 {(x - 2)}^{2} - 4 y^{2} + 24 y = 36 + 36$

Step 1.5

Complete the square for $- 4 y^{2} + 24 y$ .

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Step 1.5.1

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

$a = - 4$

$b = 24$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

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

Step 1.5.3

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

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Step 1.5.3.1

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

$d = \frac{24}{2 \cdot - 4}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $24$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $24$ .

$d = \frac{2 \cdot 12}{2 \cdot - 4}$

Step 1.5.3.2.1.2

Cancel the common factors.

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Step 1.5.3.2.1.2.1

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

$d = \frac{2 \cdot 12}{2 (- 4)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 12}{2 \cdot - 4}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{12}{- 4}$

Step 1.5.3.2.2

Cancel the common factor of $12$ and $- 4$ .

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Step 1.5.3.2.2.1

Factor $4$ out of $12$ .

$d = \frac{4 (3)}{- 4}$

Step 1.5.3.2.2.2

Move the negative one from the denominator of $\frac{3}{- 1}$ .

$d = - 1 \cdot 3$

Step 1.5.3.2.3

Multiply $- 1$ by $3$ .

$d = - 3$

Step 1.5.4

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

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Step 1.5.4.1

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

$e = 0 - \frac{24^{2}}{4 \cdot - 4}$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Raise $24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot - 4}$

Step 1.5.4.2.1.2

Multiply $4$ by $- 4$ .

$e = 0 - \frac{576}{- 16}$

Step 1.5.4.2.1.3

Divide $576$ by $- 16$ .

$e = 0 - - 36$

Step 1.5.4.2.1.4

Multiply $- 1$ by $- 36$ .

$e = 0 + 36$

Step 1.5.4.2.2

Add $0$ and $36$ .

$e = 36$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(y - 3)}^{2} + 36$ .

$- 4 {(y - 3)}^{2} + 36$

Step 1.6

Substitute $- 4 {(y - 3)}^{2} + 36$ for $- 4 y^{2} + 24 y$ in the equation $9 x^{2} - 36 x - 4 y^{2} + 24 y = 36$ .

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

Step 1.7

Move $36$ to the right side of the equation by adding $36$ to both sides.

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

Step 1.8

Simplify $36 + 36 - 36$ .

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Step 1.8.1

Add $36$ and $36$ .

$9 {(x - 2)}^{2} - 4 {(y - 3)}^{2} = 72 - 36$

Step 1.8.2

Subtract $36$ from $72$ .

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

Step 1.9

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

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

Step 1.10

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 $1$ .

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

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find the asymptotes of the hyperbola.

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

Step 3

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, $a$ .

$a = 2$

$b = 3$

$k = 3$

$h = 2$

Step 4

The asymptotes follow the form $y = \pm \frac{b (x - h)}{a} + k$ because this hyperbola opens left and right.

$y = \pm \frac{3}{2} \cdot (x - (2)) + 3$

Step 5

Simplify to find the first asymptote.

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Step 5.1

Remove parentheses.

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

Step 5.2

Simplify $\frac{3}{2} \cdot (x - (2)) + 3$ .

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $- 1$ by $2$ .

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

Step 5.2.1.2

Apply the distributive property.

$y = \frac{3}{2} x + \frac{3}{2} \cdot - 2 + 3$

Step 5.2.1.3

Combine $\frac{3}{2}$ and $x$ .

$y = \frac{3 x}{2} + \frac{3}{2} \cdot - 2 + 3$

Step 5.2.1.4

Cancel the common factor of $2$ .

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Step 5.2.1.4.1

Factor $2$ out of $- 2$ .

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

Step 5.2.1.4.2

Cancel the common factor.

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

Step 5.2.1.4.3

Rewrite the expression.

$y = \frac{3 x}{2} + 3 \cdot - 1 + 3$

Step 5.2.1.5

Multiply $3$ by $- 1$ .

$y = \frac{3 x}{2} - 3 + 3$

Step 5.2.2

Combine the opposite terms in $\frac{3 x}{2} - 3 + 3$ .

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Step 5.2.2.1

Add $- 3$ and $3$ .

$y = \frac{3 x}{2} + 0$

Step 5.2.2.2

Add $\frac{3 x}{2}$ and $0$ .

$y = \frac{3 x}{2}$

Step 6

Simplify to find the second asymptote.

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Step 6.1

Remove parentheses.

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

Step 6.2

Simplify $- \frac{3}{2} \cdot (x - (2)) + 3$ .

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $- 1$ by $2$ .

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

Step 6.2.1.2

Apply the distributive property.

$y = - \frac{3}{2} x - \frac{3}{2} \cdot - 2 + 3$

Step 6.2.1.3

Combine $x$ and $\frac{3}{2}$ .

$y = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot - 2 + 3$

Step 6.2.1.4

Cancel the common factor of $2$ .

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Step 6.2.1.4.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot - 2 + 3$

Step 6.2.1.4.2

Factor $2$ out of $- 2$ .

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

Step 6.2.1.4.3

Cancel the common factor.

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

Step 6.2.1.4.4

Rewrite the expression.

$y = - \frac{x \cdot 3}{2} - 3 \cdot - 1 + 3$

Step 6.2.1.5

Multiply $- 3$ by $- 1$ .

$y = - \frac{x \cdot 3}{2} + 3 + 3$

Step 6.2.1.6

Move $3$ to the left of $x$ .

$y = - \frac{3 x}{2} + 3 + 3$

Step 6.2.2

Add $3$ and $3$ .

$y = - \frac{3 x}{2} + 6$

Step 7

This hyperbola has two asymptotes.

$y = \frac{3 x}{2}, y = - \frac{3 x}{2} + 6$

Step 8

The asymptotes are $y = \frac{3 x}{2}$ and $y = - \frac{3 x}{2} + 6$ .

Asymptotes: $y = \frac{3 x}{2}, y = - \frac{3 x}{2} + 6$

Step 9