Precalculus Examples

$36 x^{2} + 49 y^{2} - 216 x + 98 y - 1391 = 0$

Step 1

Add $1391$ to both sides of the equation.

$36 x^{2} + 49 y^{2} - 216 x + 98 y = 1391$

Step 2

Complete the square for $36 x^{2} - 216 x$ .

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Step 2.1

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

$a = 36$

$b = - 216$

$c = 0$

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

$d = \frac{- 216}{2 \cdot 36}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 216$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 216$ .

$d = \frac{2 \cdot - 108}{2 \cdot 36}$

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 $2 \cdot 36$ .

$d = \frac{2 \cdot - 108}{2 (36)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 108}{2 \cdot 36}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 108}{36}$

Step 2.3.2.2

Cancel the common factor of $- 108$ and $36$ .

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Step 2.3.2.2.1

Factor $36$ out of $- 108$ .

$d = \frac{36 \cdot - 3}{36}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $36$ out of $36$ .

$d = \frac{36 \cdot - 3}{36 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{36 \cdot - 3}{36 \cdot 1}$

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

$e = 0 - \frac{{(- 216)}^{2}}{4 \cdot 36}$

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 $- 216$ to the power of $2$ .

$e = 0 - \frac{46656}{4 \cdot 36}$

Step 2.4.2.1.2

Multiply $4$ by $36$ .

$e = 0 - \frac{46656}{144}$

Step 2.4.2.1.3

Divide $46656$ by $144$ .

$e = 0 - 1 \cdot 324$

Step 2.4.2.1.4

Multiply $- 1$ by $324$ .

$e = 0 - 324$

Step 2.4.2.2

Subtract $324$ from $0$ .

$e = - 324$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $36 {(x - 3)}^{2} - 324$ .

$36 {(x - 3)}^{2} - 324$

Step 3

Substitute $36 {(x - 3)}^{2} - 324$ for $36 x^{2} - 216 x$ in the equation $36 x^{2} + 49 y^{2} - 216 x + 98 y = 1391$ .

$36 {(x - 3)}^{2} - 324 + 49 y^{2} + 98 y = 1391$

Step 4

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

$36 {(x - 3)}^{2} + 49 y^{2} + 98 y = 1391 + 324$

Step 5

Complete the square for $49 y^{2} + 98 y$ .

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Step 5.1

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

$a = 49$

$b = 98$

$c = 0$

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

$d = \frac{98}{2 \cdot 49}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $98$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $98$ .

$d = \frac{2 \cdot 49}{2 \cdot 49}$

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 $2 \cdot 49$ .

$d = \frac{2 \cdot 49}{2 (49)}$

Step 5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 49}{2 \cdot 49}$

Step 5.3.2.1.2.3

Rewrite the expression.

$d = \frac{49}{49}$

Step 5.3.2.2

Cancel the common factor of $49$ .

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Step 5.3.2.2.1

Cancel the common factor.

$d = \frac{49}{49}$

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

$e = 0 - \frac{98^{2}}{4 \cdot 49}$

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 $98$ to the power of $2$ .

$e = 0 - \frac{9604}{4 \cdot 49}$

Step 5.4.2.1.2

Multiply $4$ by $49$ .

$e = 0 - \frac{9604}{196}$

Step 5.4.2.1.3

Divide $9604$ by $196$ .

$e = 0 - 1 \cdot 49$

Step 5.4.2.1.4

Multiply $- 1$ by $49$ .

$e = 0 - 49$

Step 5.4.2.2

Subtract $49$ from $0$ .

$e = - 49$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $49 {(y + 1)}^{2} - 49$ .

$49 {(y + 1)}^{2} - 49$

Step 6

Substitute $49 {(y + 1)}^{2} - 49$ for $49 y^{2} + 98 y$ in the equation $36 x^{2} + 49 y^{2} - 216 x + 98 y = 1391$ .

$36 {(x - 3)}^{2} + 49 {(y + 1)}^{2} - 49 = 1391 + 324$

Step 7

Move $- 49$ to the right side of the equation by adding $49$ to both sides.

$36 {(x - 3)}^{2} + 49 {(y + 1)}^{2} = 1391 + 324 + 49$

Step 8

Simplify $1391 + 324 + 49$ .

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Step 8.1

Add $1391$ and $324$ .

$36 {(x - 3)}^{2} + 49 {(y + 1)}^{2} = 1715 + 49$

Step 8.2

Add $1715$ and $49$ .

$36 {(x - 3)}^{2} + 49 {(y + 1)}^{2} = 1764$

Step 9

Divide each term by $1764$ to make the right side equal to one.

$\frac{36 {(x - 3)}^{2}}{1764} + \frac{49 {(y + 1)}^{2}}{1764} = \frac{1764}{1764}$

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 $1$ .

$\frac{{(x - 3)}^{2}}{49} + \frac{{(y + 1)}^{2}}{36} = 1$