Precalculus Examples

$4 x^{2} + 24 x + 16 y^{2} - 32 y - 12 = 0$

Step 1

Add $12$ to both sides of the equation.

$4 x^{2} + 24 x + 16 y^{2} - 32 y = 12$

Step 2

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

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Step 2.1

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

$a = 4$

$b = 24$

$c = 0$

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Factor $2$ out of $24$ .

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

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 $2 \cdot 4$ .

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

Step 2.3.2.1.2.2

Cancel the common factor.

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

Step 2.3.2.1.2.3

Rewrite the expression.

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

Factor $4$ out of $12$ .

$d = \frac{4 \cdot 3}{4}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $4$ out of $4$ .

$d = \frac{4 \cdot 3}{4 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{4 \cdot 3}{4 \cdot 1}$

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

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

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 $24$ to the power of $2$ .

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

Step 2.4.2.1.2

Multiply $4$ by $4$ .

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

Step 2.4.2.1.3

Divide $576$ by $16$ .

$e = 0 - 1 \cdot 36$

Step 2.4.2.1.4

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Step 2.4.2.2

Subtract $36$ from $0$ .

$e = - 36$

Step 2.5

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

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

Step 3

Substitute $4 {(x + 3)}^{2} - 36$ for $4 x^{2} + 24 x$ in the equation $4 x^{2} + 24 x + 16 y^{2} - 32 y = 12$ .

$4 {(x + 3)}^{2} - 36 + 16 y^{2} - 32 y = 12$

Step 4

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

$4 {(x + 3)}^{2} + 16 y^{2} - 32 y = 12 + 36$

Step 5

Complete the square for $16 y^{2} - 32 y$ .

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Step 5.1

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

$a = 16$

$b = - 32$

$c = 0$

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

$d = \frac{- 32}{2 \cdot 16}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $- 32$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $- 32$ .

$d = \frac{2 \cdot - 16}{2 \cdot 16}$

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 $2 \cdot 16$ .

$d = \frac{2 \cdot - 16}{2 (16)}$

Step 5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 16}{2 \cdot 16}$

Step 5.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 16}{16}$

Step 5.3.2.2

Cancel the common factor of $- 16$ and $16$ .

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Step 5.3.2.2.1

Factor $16$ out of $- 16$ .

$d = \frac{16 \cdot - 1}{16}$

Step 5.3.2.2.2

Cancel the common factors.

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Step 5.3.2.2.2.1

Factor $16$ out of $16$ .

$d = \frac{16 \cdot - 1}{16 (1)}$

Step 5.3.2.2.2.2

Cancel the common factor.

$d = \frac{16 \cdot - 1}{16 \cdot 1}$

Step 5.3.2.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

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

$e = 0 - \frac{{(- 32)}^{2}}{4 \cdot 16}$

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 $- 32$ to the power of $2$ .

$e = 0 - \frac{1024}{4 \cdot 16}$

Step 5.4.2.1.2

Multiply $4$ by $16$ .

$e = 0 - \frac{1024}{64}$

Step 5.4.2.1.3

Divide $1024$ by $64$ .

$e = 0 - 1 \cdot 16$

Step 5.4.2.1.4

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Step 5.4.2.2

Subtract $16$ from $0$ .

$e = - 16$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(y - 1)}^{2} - 16$ .

$16 {(y - 1)}^{2} - 16$

Step 6

Substitute $16 {(y - 1)}^{2} - 16$ for $16 y^{2} - 32 y$ in the equation $4 x^{2} + 24 x + 16 y^{2} - 32 y = 12$ .

$4 {(x + 3)}^{2} + 16 {(y - 1)}^{2} - 16 = 12 + 36$

Step 7

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

$4 {(x + 3)}^{2} + 16 {(y - 1)}^{2} = 12 + 36 + 16$

Step 8

Simplify $12 + 36 + 16$ .

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Step 8.1

Add $12$ and $36$ .

$4 {(x + 3)}^{2} + 16 {(y - 1)}^{2} = 48 + 16$

Step 8.2

Add $48$ and $16$ .

$4 {(x + 3)}^{2} + 16 {(y - 1)}^{2} = 64$

Step 9

Divide each term by $64$ to make the right side equal to one.

$\frac{4 {(x + 3)}^{2}}{64} + \frac{16 {(y - 1)}^{2}}{64} = \frac{64}{64}$

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 $1$ .

$\frac{{(x + 3)}^{2}}{16} + \frac{{(y - 1)}^{2}}{4} = 1$