Precalculus Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 1.1.1.2

Add $4 x$ to both sides of the equation.

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

Step 1.1.1.3

Add $12$ to both sides of the equation.

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

Step 1.1.2

Divide each term in $- 16 y = - x^{2} + 4 x + 12$ by $- 16$ and simplify.

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Step 1.1.2.1

Divide each term in $- 16 y = - x^{2} + 4 x + 12$ by $- 16$ .

$\frac{- 16 y}{- 16} = \frac{- x^{2}}{- 16} + \frac{4 x}{- 16} + \frac{12}{- 16}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 16$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 16 y}{- 16} = \frac{- x^{2}}{- 16} + \frac{4 x}{- 16} + \frac{12}{- 16}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 16} + \frac{4 x}{- 16} + \frac{12}{- 16}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{16} + \frac{4 x}{- 16} + \frac{12}{- 16}$

Step 1.1.2.3.1.2

Cancel the common factor of $4$ and $- 16$ .

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Step 1.1.2.3.1.2.1

Factor $4$ out of $4 x$ .

$y = \frac{x^{2}}{16} + \frac{4 (x)}{- 16} + \frac{12}{- 16}$

Step 1.1.2.3.1.2.2

Cancel the common factors.

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Step 1.1.2.3.1.2.2.1

Factor $4$ out of $- 16$ .

$y = \frac{x^{2}}{16} + \frac{4 x}{4 \cdot - 4} + \frac{12}{- 16}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{2}}{16} + \frac{4 x}{4 \cdot - 4} + \frac{12}{- 16}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{2}}{16} + \frac{x}{- 4} + \frac{12}{- 16}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$y = \frac{x^{2}}{16} - \frac{x}{4} + \frac{12}{- 16}$

Step 1.1.2.3.1.4

Cancel the common factor of $12$ and $- 16$ .

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Step 1.1.2.3.1.4.1

Factor $4$ out of $12$ .

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

Step 1.1.2.3.1.4.2

Cancel the common factors.

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Step 1.1.2.3.1.4.2.1

Factor $4$ out of $- 16$ .

$y = \frac{x^{2}}{16} - \frac{x}{4} + \frac{4 \cdot 3}{4 \cdot - 4}$

Step 1.1.2.3.1.4.2.2

Cancel the common factor.

$y = \frac{x^{2}}{16} - \frac{x}{4} + \frac{4 \cdot 3}{4 \cdot - 4}$

Step 1.1.2.3.1.4.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.5

Move the negative in front of the fraction.

$y = \frac{x^{2}}{16} - \frac{x}{4} - \frac{3}{4}$

Step 1.2

Complete the square for $\frac{x^{2}}{16} - \frac{x}{4} - \frac{3}{4}$ .

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Step 1.2.1

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

$a = \frac{1}{16}$

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

$c = - \frac{3}{4}$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{- \frac{1}{4}}{2 (\frac{1}{16})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{4} \cdot \frac{1}{2 (\frac{1}{16})}$

Step 1.2.3.2.2

Combine $2$ and $\frac{1}{16}$ .

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2}{16}}$

Step 1.2.3.2.3

Cancel the common factor of $2$ and $16$ .

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Step 1.2.3.2.3.1

Factor $2$ out of $2$ .

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2 (1)}{16}}$

Step 1.2.3.2.3.2

Cancel the common factors.

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Step 1.2.3.2.3.2.1

Factor $2$ out of $16$ .

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Step 1.2.3.2.3.2.2

Cancel the common factor.

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Step 1.2.3.2.3.2.3

Rewrite the expression.

$d = - \frac{1}{4} \cdot \frac{1}{\frac{1}{8}}$

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{4} (1 \cdot 8)$

Step 1.2.3.2.5

Multiply $8$ by $1$ .

$d = - \frac{1}{4} \cdot 8$

Step 1.2.3.2.6

Cancel the common factor of $4$ .

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Step 1.2.3.2.6.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$d = \frac{- 1}{4} \cdot 8$

Step 1.2.3.2.6.2

Factor $4$ out of $8$ .

$d = \frac{- 1}{4} \cdot (4 (2))$

Step 1.2.3.2.6.3

Cancel the common factor.

$d = \frac{- 1}{4} \cdot (4 \cdot 2)$

Step 1.2.3.2.6.4

Rewrite the expression.

$d = - 1 \cdot 2$

Step 1.2.3.2.7

Multiply $- 1$ by $2$ .

$d = - 2$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = - \frac{3}{4} - \frac{{(- \frac{1}{4})}^{2}}{4 (\frac{1}{16})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

$e = - \frac{3}{4} - \frac{{(- 1)}^{2} {(\frac{1}{4})}^{2}}{4 (\frac{1}{16})}$

Step 1.2.4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$e = - \frac{3}{4} - \frac{1 {(\frac{1}{4})}^{2}}{4 (\frac{1}{16})}$

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{1}{4}$ .

$e = - \frac{3}{4} - \frac{1 \frac{1^{2}}{4^{2}}}{4 (\frac{1}{16})}$

Step 1.2.4.2.1.1.4

One to any power is one.

$e = - \frac{3}{4} - \frac{1 \frac{1}{4^{2}}}{4 (\frac{1}{16})}$

Step 1.2.4.2.1.1.5

Raise $4$ to the power of $2$ .

$e = - \frac{3}{4} - \frac{1 (\frac{1}{16})}{4 (\frac{1}{16})}$

Step 1.2.4.2.1.1.6

Multiply $\frac{1}{16}$ by $1$ .

$e = - \frac{3}{4} - \frac{\frac{1}{16}}{4 (\frac{1}{16})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{16}$ .

$e = - \frac{3}{4} - \frac{\frac{1}{16}}{\frac{4}{16}}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ and $16$ .

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Step 1.2.4.2.1.3.1

Factor $4$ out of $4$ .

$e = - \frac{3}{4} - \frac{\frac{1}{16}}{\frac{4 (1)}{16}}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor $4$ out of $16$ .

$e = - \frac{3}{4} - \frac{\frac{1}{16}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = - \frac{3}{4} - \frac{\frac{1}{16}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = - \frac{3}{4} - \frac{\frac{1}{16}}{\frac{1}{4}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{3}{4} - (\frac{1}{16} \cdot 4)$

Step 1.2.4.2.1.5

Cancel the common factor of $4$ .

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Step 1.2.4.2.1.5.1

Factor $4$ out of $16$ .

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

Step 1.2.4.2.1.5.2

Cancel the common factor.

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

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = - \frac{3}{4} - \frac{1}{4}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 3 - 1}{4}$

Step 1.2.4.2.3

Subtract $1$ from $- 3$ .

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

Step 1.2.4.2.4

Divide $- 4$ by $4$ .

$e = - 1$

Step 1.2.5

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

$\frac{1}{16} {(x - 2)}^{2} - 1$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = \frac{1}{16}$

$h = 2$

$k = - 1$

Step 3

Find the vertex $(h, k)$ .

$(2, - 1)$

Step 4