Precalculus Examples

$16 x^{2} + 288 y + 36 y^{2} = 0$

Step 1

Complete the square for $288 y + 36 y^{2}$ .

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Step 1.1

Reorder $288 y$ and $36 y^{2}$ .

$36 y^{2} + 288 y$

Step 1.2

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

$a = 36$

$b = 288$

$c = 0$

Step 1.3

Consider the vertex form of a parabola.

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

Step 1.4

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

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Step 1.4.1

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

$d = \frac{288}{2 \cdot 36}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Cancel the common factor of $288$ and $2$ .

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Step 1.4.2.1.1

Factor $2$ out of $288$ .

$d = \frac{2 \cdot 144}{2 \cdot 36}$

Step 1.4.2.1.2

Cancel the common factors.

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Step 1.4.2.1.2.1

Factor $2$ out of $2 \cdot 36$ .

$d = \frac{2 \cdot 144}{2 (36)}$

Step 1.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 144}{2 \cdot 36}$

Step 1.4.2.1.2.3

Rewrite the expression.

$d = \frac{144}{36}$

Step 1.4.2.2

Cancel the common factor of $144$ and $36$ .

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Step 1.4.2.2.1

Factor $36$ out of $144$ .

$d = \frac{36 \cdot 4}{36}$

Step 1.4.2.2.2

Cancel the common factors.

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Step 1.4.2.2.2.1

Factor $36$ out of $36$ .

$d = \frac{36 \cdot 4}{36 (1)}$

Step 1.4.2.2.2.2

Cancel the common factor.

$d = \frac{36 \cdot 4}{36 \cdot 1}$

Step 1.4.2.2.2.3

Rewrite the expression.

$d = \frac{4}{1}$

Step 1.4.2.2.2.4

Divide $4$ by $1$ .

$d = 4$

Step 1.5

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

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Step 1.5.1

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

$e = 0 - \frac{288^{2}}{4 \cdot 36}$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

Raise $288$ to the power of $2$ .

$e = 0 - \frac{82944}{4 \cdot 36}$

Step 1.5.2.1.2

Multiply $4$ by $36$ .

$e = 0 - \frac{82944}{144}$

Step 1.5.2.1.3

Divide $82944$ by $144$ .

$e = 0 - 1 \cdot 576$

Step 1.5.2.1.4

Multiply $- 1$ by $576$ .

$e = 0 - 576$

Step 1.5.2.2

Subtract $576$ from $0$ .

$e = - 576$

Step 1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $36 {(y + 4)}^{2} - 576$ .

$36 {(y + 4)}^{2} - 576$

Step 2

Substitute $36 {(y + 4)}^{2} - 576$ for $288 y + 36 y^{2}$ in the equation $16 x^{2} + 288 y + 36 y^{2} = 0$ .

$16 x^{2} + 36 {(y + 4)}^{2} - 576 = 0$

Step 3

Move $- 576$ to the right side of the equation by adding $576$ to both sides.

$16 x^{2} + 36 {(y + 4)}^{2} = 0 + 576$

Step 4

Add $0$ and $576$ .

$16 x^{2} + 36 {(y + 4)}^{2} = 576$

Step 5

Divide each term by $576$ to make the right side equal to one.

$\frac{16 x^{2}}{576} + \frac{36 {(y + 4)}^{2}}{576} = \frac{576}{576}$

Step 6

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 $1$ .

$\frac{x^{2}}{36} + \frac{{(y + 4)}^{2}}{16} = 1$