Precalculus Examples

$x^{2} - 12 y - 6 x + 5 = 0$

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.

$- 12 y - 6 x + 5 = - x^{2}$

Step 1.1.1.2

Add $6 x$ to both sides of the equation.

$- 12 y + 5 = - x^{2} + 6 x$

Step 1.1.1.3

Subtract $5$ from both sides of the equation.

$- 12 y = - x^{2} + 6 x - 5$

Step 1.1.2

Divide each term in $- 12 y = - x^{2} + 6 x - 5$ by $- 12$ and simplify.

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Step 1.1.2.1

Divide each term in $- 12 y = - x^{2} + 6 x - 5$ by $- 12$ .

$\frac{- 12 y}{- 12} = \frac{- x^{2}}{- 12} + \frac{6 x}{- 12} + \frac{- 5}{- 12}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 12$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 12 y}{- 12} = \frac{- x^{2}}{- 12} + \frac{6 x}{- 12} + \frac{- 5}{- 12}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 12} + \frac{6 x}{- 12} + \frac{- 5}{- 12}$

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.

$y = \frac{x^{2}}{12} + \frac{6 x}{- 12} + \frac{- 5}{- 12}$

Step 1.1.2.3.1.2

Cancel the common factor of $6$ and $- 12$ .

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Step 1.1.2.3.1.2.1

Factor $6$ out of $6 x$ .

$y = \frac{x^{2}}{12} + \frac{6 (x)}{- 12} + \frac{- 5}{- 12}$

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 $6$ out of $- 12$ .

$y = \frac{x^{2}}{12} + \frac{6 x}{6 \cdot - 2} + \frac{- 5}{- 12}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{2}}{12} + \frac{6 x}{6 \cdot - 2} + \frac{- 5}{- 12}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{2}}{12} + \frac{x}{- 2} + \frac{- 5}{- 12}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$y = \frac{x^{2}}{12} - \frac{x}{2} + \frac{- 5}{- 12}$

Step 1.1.2.3.1.4

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{12} - \frac{x}{2} + \frac{5}{12}$

Step 1.2

Complete the square for $\frac{x^{2}}{12} - \frac{x}{2} + \frac{5}{12}$ .

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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 $c$ .

$a = \frac{1}{12}$

$b = - \frac{1}{2}$

$c = \frac{5}{12}$

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

$d = \frac{- \frac{1}{2}}{2 (\frac{1}{12})}$

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.

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

Step 1.2.3.2.2

Combine $2$ and $\frac{1}{12}$ .

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

Step 1.2.3.2.3

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

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Step 1.2.3.2.3.1

Factor $2$ out of $2$ .

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

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 $12$ .

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

Step 1.2.3.2.3.2.2

Cancel the common factor.

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

Step 1.2.3.2.3.2.3

Rewrite the expression.

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

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.5

Multiply $6$ by $1$ .

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

Step 1.2.3.2.6

Cancel the common factor of $2$ .

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Step 1.2.3.2.6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.2.3.2.6.2

Factor $2$ out of $6$ .

$d = \frac{- 1}{2} \cdot (2 (3))$

Step 1.2.3.2.6.3

Cancel the common factor.

$d = \frac{- 1}{2} \cdot (2 \cdot 3)$

Step 1.2.3.2.6.4

Rewrite the expression.

$d = - 1 \cdot 3$

Step 1.2.3.2.7

Multiply $- 1$ by $3$ .

$d = - 3$

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

$e = \frac{5}{12} - \frac{{(- \frac{1}{2})}^{2}}{4 (\frac{1}{12})}$

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 $- \frac{1}{2}$ .

$e = \frac{5}{12} - \frac{{(- 1)}^{2} {(\frac{1}{2})}^{2}}{4 (\frac{1}{12})}$

Step 1.2.4.2.1.1.2

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

$e = \frac{5}{12} - \frac{1 {(\frac{1}{2})}^{2}}{4 (\frac{1}{12})}$

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{1}{2}$ .

$e = \frac{5}{12} - \frac{1 \frac{1^{2}}{2^{2}}}{4 (\frac{1}{12})}$

Step 1.2.4.2.1.1.4

One to any power is one.

$e = \frac{5}{12} - \frac{1 \frac{1}{2^{2}}}{4 (\frac{1}{12})}$

Step 1.2.4.2.1.1.5

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

$e = \frac{5}{12} - \frac{1 (\frac{1}{4})}{4 (\frac{1}{12})}$

Step 1.2.4.2.1.1.6

Multiply $\frac{1}{4}$ by $1$ .

$e = \frac{5}{12} - \frac{\frac{1}{4}}{4 (\frac{1}{12})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{12}$ .

$e = \frac{5}{12} - \frac{\frac{1}{4}}{\frac{4}{12}}$

Step 1.2.4.2.1.3

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

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Step 1.2.4.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{5}{12} - \frac{\frac{1}{4}}{\frac{4 (1)}{12}}$

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 $12$ .

$e = \frac{5}{12} - \frac{\frac{1}{4}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = \frac{5}{12} - \frac{\frac{1}{4}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = \frac{5}{12} - \frac{\frac{1}{4}}{\frac{1}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{5}{12} - (\frac{1}{4} \cdot 3)$

Step 1.2.4.2.1.5

Combine $\frac{1}{4}$ and $3$ .

$e = \frac{5}{12} - \frac{3}{4}$

Step 1.2.4.2.2

To write $- \frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$e = \frac{5}{12} - \frac{3}{4} \cdot \frac{3}{3}$

Step 1.2.4.2.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.4.2.3.1

Multiply $\frac{3}{4}$ by $\frac{3}{3}$ .

$e = \frac{5}{12} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 1.2.4.2.3.2

Multiply $4$ by $3$ .

$e = \frac{5}{12} - \frac{3 \cdot 3}{12}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{5 - 3 \cdot 3}{12}$

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $- 3$ by $3$ .

$e = \frac{5 - 9}{12}$

Step 1.2.4.2.5.2

Subtract $9$ from $5$ .

$e = \frac{- 4}{12}$

Step 1.2.4.2.6

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

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Step 1.2.4.2.6.1

Factor $4$ out of $- 4$ .

$e = \frac{4 (- 1)}{12}$

Step 1.2.4.2.6.2

Cancel the common factors.

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Step 1.2.4.2.6.2.1

Factor $4$ out of $12$ .

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

Step 1.2.4.2.6.2.2

Cancel the common factor.

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

Step 1.2.4.2.6.2.3

Rewrite the expression.

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

Step 1.2.4.2.7

Move the negative in front of the fraction.

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

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{12} {(x - 3)}^{2} - \frac{1}{3}$ .

$\frac{1}{12} {(x - 3)}^{2} - \frac{1}{3}$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{12} \cdot {(x - 3)}^{2} - \frac{1}{3}$

Step 2

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

$a = \frac{1}{12}$

$h = 3$

$k = - \frac{1}{3}$

Step 3

Find the vertex $(h, k)$ .

$(3, - \frac{1}{3})$

Step 4