Precalculus Examples

$9 y^{2} + 108 y + 4 x^{2} - 56 x = - 484$

Step 1

Complete the square for $9 y^{2} + 108 y$ .

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Step 1.1

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

$a = 9$

$b = 108$

$c = 0$

Step 1.2

Consider the vertex form of a parabola.

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

Step 1.3

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

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Step 1.3.1

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

$d = \frac{108}{2 \cdot 9}$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

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

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Step 1.3.2.1.1

Factor $2$ out of $108$ .

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

Step 1.3.2.1.2

Cancel the common factors.

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Step 1.3.2.1.2.1

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

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

Step 1.3.2.1.2.2

Cancel the common factor.

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

Step 1.3.2.1.2.3

Rewrite the expression.

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

Step 1.3.2.2

Cancel the common factor of $54$ and $9$ .

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Step 1.3.2.2.1

Factor $9$ out of $54$ .

$d = \frac{9 \cdot 6}{9}$

Step 1.3.2.2.2

Cancel the common factors.

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Step 1.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot 6}{9 (1)}$

Step 1.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot 6}{9 \cdot 1}$

Step 1.3.2.2.2.3

Rewrite the expression.

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

Step 1.3.2.2.2.4

Divide $6$ by $1$ .

$d = 6$

Step 1.4

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

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Step 1.4.1

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

$e = 0 - \frac{108^{2}}{4 \cdot 9}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Raise $108$ to the power of $2$ .

$e = 0 - \frac{11664}{4 \cdot 9}$

Step 1.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{11664}{36}$

Step 1.4.2.1.3

Divide $11664$ by $36$ .

$e = 0 - 1 \cdot 324$

Step 1.4.2.1.4

Multiply $- 1$ by $324$ .

$e = 0 - 324$

Step 1.4.2.2

Subtract $324$ from $0$ .

$e = - 324$

Step 1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(y + 6)}^{2} - 324$ .

$9 {(y + 6)}^{2} - 324$

Step 2

Substitute $9 {(y + 6)}^{2} - 324$ for $9 y^{2} + 108 y$ in the equation $9 y^{2} + 108 y + 4 x^{2} - 56 x = - 484$ .

$9 {(y + 6)}^{2} - 324 + 4 x^{2} - 56 x = - 484$

Step 3

Move $- 324$ to the right side of the equation by adding $324$ to both sides.

$9 {(y + 6)}^{2} + 4 x^{2} - 56 x = - 484 + 324$

Step 4

Complete the square for $4 x^{2} - 56 x$ .

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Step 4.1

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

$a = 4$

$b = - 56$

$c = 0$

Step 4.2

Consider the vertex form of a parabola.

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

Step 4.3

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

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Step 4.3.1

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

$d = \frac{- 56}{2 \cdot 4}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Cancel the common factor of $- 56$ and $2$ .

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Step 4.3.2.1.1

Factor $2$ out of $- 56$ .

$d = \frac{2 \cdot - 28}{2 \cdot 4}$

Step 4.3.2.1.2

Cancel the common factors.

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Step 4.3.2.1.2.1

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

$d = \frac{2 \cdot - 28}{2 (4)}$

Step 4.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 28}{2 \cdot 4}$

Step 4.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 28}{4}$

Step 4.3.2.2

Cancel the common factor of $- 28$ and $4$ .

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Step 4.3.2.2.1

Factor $4$ out of $- 28$ .

$d = \frac{4 \cdot - 7}{4}$

Step 4.3.2.2.2

Cancel the common factors.

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Step 4.3.2.2.2.1

Factor $4$ out of $4$ .

$d = \frac{4 \cdot - 7}{4 (1)}$

Step 4.3.2.2.2.2

Cancel the common factor.

$d = \frac{4 \cdot - 7}{4 \cdot 1}$

Step 4.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 7}{1}$

Step 4.3.2.2.2.4

Divide $- 7$ by $1$ .

$d = - 7$

Step 4.4

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

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Step 4.4.1

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

$e = 0 - \frac{{(- 56)}^{2}}{4 \cdot 4}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Raise $- 56$ to the power of $2$ .

$e = 0 - \frac{3136}{4 \cdot 4}$

Step 4.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{3136}{16}$

Step 4.4.2.1.3

Divide $3136$ by $16$ .

$e = 0 - 1 \cdot 196$

Step 4.4.2.1.4

Multiply $- 1$ by $196$ .

$e = 0 - 196$

Step 4.4.2.2

Subtract $196$ from $0$ .

$e = - 196$

Step 4.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - 7)}^{2} - 196$ .

$4 {(x - 7)}^{2} - 196$

Step 5

Substitute $4 {(x - 7)}^{2} - 196$ for $4 x^{2} - 56 x$ in the equation $9 y^{2} + 108 y + 4 x^{2} - 56 x = - 484$ .

$9 {(y + 6)}^{2} + 4 {(x - 7)}^{2} - 196 = - 484 + 324$

Step 6

Move $- 196$ to the right side of the equation by adding $196$ to both sides.

$9 {(y + 6)}^{2} + 4 {(x - 7)}^{2} = - 484 + 324 + 196$

Step 7

Simplify $- 484 + 324 + 196$ .

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Step 7.1

Add $- 484$ and $324$ .

$9 {(y + 6)}^{2} + 4 {(x - 7)}^{2} = - 160 + 196$

Step 7.2

Add $- 160$ and $196$ .

$9 {(y + 6)}^{2} + 4 {(x - 7)}^{2} = 36$

Step 8

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

$\frac{9 {(y + 6)}^{2}}{36} + \frac{4 {(x - 7)}^{2}}{36} = \frac{36}{36}$

Step 9

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 $1$ .

$\frac{{(y + 6)}^{2}}{4} + \frac{{(x - 7)}^{2}}{9} = 1$