Precalculus Examples

$y^{2} - 4 x + 6 y + 29 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

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Step 1.1.1.1

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

$- 4 x + 6 y + 29 = - y^{2}$

Step 1.1.1.2

Subtract $6 y$ from both sides of the equation.

$- 4 x + 29 = - y^{2} - 6 y$

Step 1.1.1.3

Subtract $29$ from both sides of the equation.

$- 4 x = - y^{2} - 6 y - 29$

Step 1.1.2

Divide each term in $- 4 x = - y^{2} - 6 y - 29$ by $- 4$ and simplify.

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Step 1.1.2.1

Divide each term in $- 4 x = - y^{2} - 6 y - 29$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- y^{2}}{- 4} + \frac{- 6 y}{- 4} + \frac{- 29}{- 4}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- y^{2}}{- 4} + \frac{- 6 y}{- 4} + \frac{- 29}{- 4}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{- 4} + \frac{- 6 y}{- 4} + \frac{- 29}{- 4}$

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.

$x = \frac{y^{2}}{4} + \frac{- 6 y}{- 4} + \frac{- 29}{- 4}$

Step 1.1.2.3.1.2

Cancel the common factor of $- 6$ and $- 4$ .

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Step 1.1.2.3.1.2.1

Factor $- 2$ out of $- 6 y$ .

$x = \frac{y^{2}}{4} + \frac{- 2 (3 y)}{- 4} + \frac{- 29}{- 4}$

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 $- 2$ out of $- 4$ .

$x = \frac{y^{2}}{4} + \frac{- 2 (3 y)}{- 2 (2)} + \frac{- 29}{- 4}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$x = \frac{y^{2}}{4} + \frac{- 2 (3 y)}{- 2 \cdot 2} + \frac{- 29}{- 4}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$x = \frac{y^{2}}{4} + \frac{3 y}{2} + \frac{- 29}{- 4}$

Step 1.1.2.3.1.3

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{4} + \frac{3 y}{2} + \frac{29}{4}$

Step 1.2

Complete the square for $\frac{y^{2}}{4} + \frac{3 y}{2} + \frac{29}{4}$ .

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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 $c$ .

$a = \frac{1}{4}$

$b = \frac{3}{2}$

$c = \frac{29}{4}$

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

$d = \frac{\frac{3}{2}}{2 (\frac{1}{4})}$

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.

$d = \frac{3}{2} \cdot \frac{1}{2 (\frac{1}{4})}$

Step 1.2.3.2.2

Combine $2$ and $\frac{1}{4}$ .

$d = \frac{3}{2} \cdot \frac{1}{\frac{2}{4}}$

Step 1.2.3.2.3

Cancel the common factor of $2$ and $4$ .

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Step 1.2.3.2.3.1

Factor $2$ out of $2$ .

$d = \frac{3}{2} \cdot \frac{1}{\frac{2 (1)}{4}}$

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 $4$ .

$d = \frac{3}{2} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 1.2.3.2.3.2.2

Cancel the common factor.

$d = \frac{3}{2} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 1.2.3.2.3.2.3

Rewrite the expression.

$d = \frac{3}{2} \cdot \frac{1}{\frac{1}{2}}$

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.5

Cancel the common factor of $2$ .

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Step 1.2.3.2.5.1

Factor $2$ out of $1 \cdot 2$ .

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

Step 1.2.3.2.5.2

Cancel the common factor.

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

Step 1.2.3.2.5.3

Rewrite the expression.

$d = 3$

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

$e = \frac{29}{4} - \frac{{(\frac{3}{2})}^{2}}{4 (\frac{1}{4})}$

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 $\frac{3}{2}$ .

$e = \frac{29}{4} - \frac{\frac{3^{2}}{2^{2}}}{4 (\frac{1}{4})}$

Step 1.2.4.2.1.1.2

Raise $3$ to the power of $2$ .

$e = \frac{29}{4} - \frac{\frac{9}{2^{2}}}{4 (\frac{1}{4})}$

Step 1.2.4.2.1.1.3

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

$e = \frac{29}{4} - \frac{\frac{9}{4}}{4 (\frac{1}{4})}$

Step 1.2.4.2.1.2

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

$e = \frac{29}{4} - \frac{\frac{9}{4}}{\frac{4}{4}}$

Step 1.2.4.2.1.3

Divide $4$ by $4$ .

$e = \frac{29}{4} - \frac{\frac{9}{4}}{1}$

Step 1.2.4.2.1.4

Divide $\frac{9}{4}$ by $1$ .

$e = \frac{29}{4} - \frac{9}{4}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{29 - 9}{4}$

Step 1.2.4.2.3

Subtract $9$ from $29$ .

$e = \frac{20}{4}$

Step 1.2.4.2.4

Divide $20$ by $4$ .

$e = 5$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(y + 3)}^{2} + 5$ .

$\frac{1}{4} {(y + 3)}^{2} + 5$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{4} \cdot {(y + 3)}^{2} + 5$

Step 2

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

$a = \frac{1}{4}$

$h = 5$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(5, - 3)$

Step 4