Precalculus Examples

$16 x^{2} - y^{2} - 128 x - 12 y + 224 = 0$

Step 1

Subtract $224$ from both sides of the equation.

$16 x^{2} - y^{2} - 128 x - 12 y = - 224$

Step 2

Complete the square for $16 x^{2} - 128 x$ .

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Step 2.1

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

$a = 16$

$b = - 128$

$c = 0$

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

$d = \frac{- 128}{2 \cdot 16}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 128$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 128$ .

$d = \frac{2 \cdot - 64}{2 \cdot 16}$

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 $2 \cdot 16$ .

$d = \frac{2 \cdot - 64}{2 (16)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 64}{2 \cdot 16}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 64}{16}$

Step 2.3.2.2

Cancel the common factor of $- 64$ and $16$ .

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Step 2.3.2.2.1

Factor $16$ out of $- 64$ .

$d = \frac{16 \cdot - 4}{16}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $16$ out of $16$ .

$d = \frac{16 \cdot - 4}{16 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{16 \cdot - 4}{16 \cdot 1}$

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

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

$e = 0 - \frac{{(- 128)}^{2}}{4 \cdot 16}$

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 $- 128$ to the power of $2$ .

$e = 0 - \frac{16384}{4 \cdot 16}$

Step 2.4.2.1.2

Multiply $4$ by $16$ .

$e = 0 - \frac{16384}{64}$

Step 2.4.2.1.3

Divide $16384$ by $64$ .

$e = 0 - 1 \cdot 256$

Step 2.4.2.1.4

Multiply $- 1$ by $256$ .

$e = 0 - 256$

Step 2.4.2.2

Subtract $256$ from $0$ .

$e = - 256$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x - 4)}^{2} - 256$ .

$16 {(x - 4)}^{2} - 256$

Step 3

Substitute $16 {(x - 4)}^{2} - 256$ for $16 x^{2} - 128 x$ in the equation $16 x^{2} - y^{2} - 128 x - 12 y = - 224$ .

$16 {(x - 4)}^{2} - 256 - y^{2} - 12 y = - 224$

Step 4

Move $- 256$ to the right side of the equation by adding $256$ to both sides.

$16 {(x - 4)}^{2} - y^{2} - 12 y = - 224 + 256$

Step 5

Complete the square for $- y^{2} - 12 y$ .

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Step 5.1

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

$a = - 1$

$b = - 12$

$c = 0$

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

$d = \frac{- 12}{2 \cdot - 1}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $- 12$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot - 1}$

Step 5.3.2.1.2

Move the negative one from the denominator of $\frac{- 6}{- 1}$ .

$d = - 1 \cdot - 6$

Step 5.3.2.2

Rewrite $- 1 \cdot - 6$ as $- - 6$ .

$d = - - 6$

Step 5.3.2.3

Multiply $- 1$ by $- 6$ .

$d = 6$

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

$e = 0 - \frac{{(- 12)}^{2}}{4 \cdot - 1}$

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 $- 12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot - 1}$

Step 5.4.2.1.2

Multiply $4$ by $- 1$ .

$e = 0 - \frac{144}{- 4}$

Step 5.4.2.1.3

Divide $144$ by $- 4$ .

$e = 0 - - 36$

Step 5.4.2.1.4

Multiply $- 1$ by $- 36$ .

$e = 0 + 36$

Step 5.4.2.2

Add $0$ and $36$ .

$e = 36$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(y + 6)}^{2} + 36$ .

$- {(y + 6)}^{2} + 36$

Step 6

Substitute $- {(y + 6)}^{2} + 36$ for $- y^{2} - 12 y$ in the equation $16 x^{2} - y^{2} - 128 x - 12 y = - 224$ .

$16 {(x - 4)}^{2} - {(y + 6)}^{2} + 36 = - 224 + 256$

Step 7

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

$16 {(x - 4)}^{2} - {(y + 6)}^{2} = - 224 + 256 - 36$

Step 8

Simplify $- 224 + 256 - 36$ .

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Step 8.1

Add $- 224$ and $256$ .

$16 {(x - 4)}^{2} - {(y + 6)}^{2} = 32 - 36$

Step 8.2

Subtract $36$ from $32$ .

$16 {(x - 4)}^{2} - {(y + 6)}^{2} = - 4$

Step 9

Flip the sign on each term of the equation so the term on the right side is positive.

$- 16 {(x - 4)}^{2} + {(y + 6)}^{2} = 4$

Step 10

Divide each term by $4$ to make the right side equal to one.

$- \frac{16 {(x - 4)}^{2}}{4} + \frac{{(y + 6)}^{2}}{4} = \frac{4}{4}$

Step 11

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 $1$ .

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