Precalculus Examples

$16 x^{2} - y^{2} + 64 x - 2 y + 67 = 0$

Step 1

Subtract $67$ from both sides of the equation.

$16 x^{2} - y^{2} + 64 x - 2 y = - 67$

Step 2

Complete the square for $16 x^{2} + 64 x$ .

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Step 2.1

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

$a = 16$

$b = 64$

$c = 0$

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $64$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $64$ .

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

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 $2 \cdot 16$ .

$d = \frac{2 \cdot 32}{2 (16)}$

Step 2.3.2.1.2.2

Cancel the common factor.

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

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{32}{16}$

Step 2.3.2.2

Cancel the common factor of $32$ and $16$ .

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Step 2.3.2.2.1

Factor $16$ out of $32$ .

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

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 $16$ .

$d = \frac{16 \cdot 2}{16 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{16 \cdot 2}{16 \cdot 1}$

Step 2.3.2.2.2.3

Rewrite the expression.

$d = \frac{2}{1}$

Step 2.3.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

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

$e = 0 - \frac{64^{2}}{4 \cdot 16}$

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 $64$ to the power of $2$ .

$e = 0 - \frac{4096}{4 \cdot 16}$

Step 2.4.2.1.2

Multiply $4$ by $16$ .

$e = 0 - \frac{4096}{64}$

Step 2.4.2.1.3

Divide $4096$ by $64$ .

$e = 0 - 1 \cdot 64$

Step 2.4.2.1.4

Multiply $- 1$ by $64$ .

$e = 0 - 64$

Step 2.4.2.2

Subtract $64$ from $0$ .

$e = - 64$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x + 2)}^{2} - 64$ .

$16 {(x + 2)}^{2} - 64$

Step 3

Substitute $16 {(x + 2)}^{2} - 64$ for $16 x^{2} + 64 x$ in the equation $16 x^{2} - y^{2} + 64 x - 2 y = - 67$ .

$16 {(x + 2)}^{2} - 64 - y^{2} - 2 y = - 67$

Step 4

Move $- 64$ to the right side of the equation by adding $64$ to both sides.

$16 {(x + 2)}^{2} - y^{2} - 2 y = - 67 + 64$

Step 5

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

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Step 5.1

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

$a = - 1$

$b = - 2$

$c = 0$

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

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

Factor $2$ out of $- 2$ .

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

Step 5.3.2.1.2

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

$d = - 1 \cdot - 1$

Step 5.3.2.2

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

$d = - - 1$

Step 5.3.2.3

Multiply $- 1$ 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 $e = c - \frac{b^{2}}{4 a}$ .

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

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

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

Step 5.4.2.1.2

Multiply $4$ by $- 1$ .

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

Step 5.4.2.1.3

Divide $4$ by $- 4$ .

$e = 0 - - 1$

Step 5.4.2.1.4

Multiply $- 1$ by $- 1$ .

$e = 0 + 1$

Step 5.4.2.2

Add $0$ and $1$ .

$e = 1$

Step 5.5

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

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

Step 6

Substitute $- {(y + 1)}^{2} + 1$ for $- y^{2} - 2 y$ in the equation $16 x^{2} - y^{2} + 64 x - 2 y = - 67$ .

$16 {(x + 2)}^{2} - {(y + 1)}^{2} + 1 = - 67 + 64$

Step 7

Move $1$ to the right side of the equation by adding $1$ to both sides.

$16 {(x + 2)}^{2} - {(y + 1)}^{2} = - 67 + 64 - 1$

Step 8

Simplify $- 67 + 64 - 1$ .

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Step 8.1

Add $- 67$ and $64$ .

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

Step 8.2

Subtract $1$ from $- 3$ .

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

Step 9

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

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

Step 10

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

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

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 $1$ .

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