Precalculus Examples

$4 y^{2} + 4 y - 16 x + 13 = 0$

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 $4 y^{2}$ from both sides of the equation.

$4 y - 16 x + 13 = - 4 y^{2}$

Step 1.1.1.2

Subtract $4 y$ from both sides of the equation.

$- 16 x + 13 = - 4 y^{2} - 4 y$

Step 1.1.1.3

Subtract $13$ from both sides of the equation.

$- 16 x = - 4 y^{2} - 4 y - 13$

Step 1.1.2

Divide each term in $- 16 x = - 4 y^{2} - 4 y - 13$ by $- 16$ and simplify.

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Step 1.1.2.1

Divide each term in $- 16 x = - 4 y^{2} - 4 y - 13$ by $- 16$ .

$\frac{- 16 x}{- 16} = \frac{- 4 y^{2}}{- 16} + \frac{- 4 y}{- 16} + \frac{- 13}{- 16}$

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.

$\frac{- 16 x}{- 16} = \frac{- 4 y^{2}}{- 16} + \frac{- 4 y}{- 16} + \frac{- 13}{- 16}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4 y^{2}}{- 16} + \frac{- 4 y}{- 16} + \frac{- 13}{- 16}$

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

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

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Step 1.1.2.3.1.1.1

Factor $- 4$ out of $- 4 y^{2}$ .

$x = \frac{- 4 (y^{2})}{- 16} + \frac{- 4 y}{- 16} + \frac{- 13}{- 16}$

Step 1.1.2.3.1.1.2

Cancel the common factors.

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Step 1.1.2.3.1.1.2.1

Factor $- 4$ out of $- 16$ .

$x = \frac{- 4 (y^{2})}{- 4 (4)} + \frac{- 4 y}{- 16} + \frac{- 13}{- 16}$

Step 1.1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{- 4 y^{2}}{- 4 \cdot 4} + \frac{- 4 y}{- 16} + \frac{- 13}{- 16}$

Step 1.1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{y^{2}}{4} + \frac{- 4 y}{- 16} + \frac{- 13}{- 16}$

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 $- 4 y$ .

$x = \frac{y^{2}}{4} + \frac{- 4 (y)}{- 16} + \frac{- 13}{- 16}$

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 $- 16$ .

$x = \frac{y^{2}}{4} + \frac{- 4 (y)}{- 4 (4)} + \frac{- 13}{- 16}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$x = \frac{y^{2}}{4} + \frac{- 4 y}{- 4 \cdot 4} + \frac{- 13}{- 16}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$x = \frac{y^{2}}{4} + \frac{y}{4} + \frac{- 13}{- 16}$

Step 1.1.2.3.1.3

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{4} + \frac{y}{4} + \frac{13}{16}$

Step 1.2

Complete the square for $\frac{y^{2}}{4} + \frac{y}{4} + \frac{13}{16}$ .

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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 $c$ .

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

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

$c = \frac{13}{16}$

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

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

Step 1.2.3.2

Cancel the common factor of $\frac{1}{4}$ .

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Step 1.2.3.2.1

Cancel the common factor.

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

Step 1.2.3.2.2

Rewrite the expression.

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

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

$e = \frac{13}{16} - \frac{{(\frac{1}{4})}^{2}}{4 (\frac{1}{4})}$

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

Cancel the common factor of ${(\frac{1}{4})}^{2}$ and $\frac{1}{4}$ .

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Step 1.2.4.2.1.1.1

Factor $\frac{1}{4}$ out of ${(\frac{1}{4})}^{2}$ .

$e = \frac{13}{16} - \frac{\frac{1}{4} \cdot \frac{1}{4}}{4 (\frac{1}{4})}$

Step 1.2.4.2.1.1.2

Cancel the common factors.

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Step 1.2.4.2.1.1.2.1

Factor $\frac{1}{4}$ out of $4 (\frac{1}{4})$ .

$e = \frac{13}{16} - \frac{\frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{4} \cdot 4}$

Step 1.2.4.2.1.1.2.2

Cancel the common factor.

$e = \frac{13}{16} - \frac{\frac{1}{4} \cdot \frac{1}{4}}{\frac{1}{4} \cdot 4}$

Step 1.2.4.2.1.1.2.3

Rewrite the expression.

$e = \frac{13}{16} - \frac{\frac{1}{4}}{4}$

Step 1.2.4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{13}{16} - (\frac{1}{4} \cdot \frac{1}{4})$

Step 1.2.4.2.1.3

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

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Step 1.2.4.2.1.3.1

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

$e = \frac{13}{16} - \frac{1}{4 \cdot 4}$

Step 1.2.4.2.1.3.2

Multiply $4$ by $4$ .

$e = \frac{13}{16} - \frac{1}{16}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{13 - 1}{16}$

Step 1.2.4.2.3

Subtract $1$ from $13$ .

$e = \frac{12}{16}$

Step 1.2.4.2.4

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

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Step 1.2.4.2.4.1

Factor $4$ out of $12$ .

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

Step 1.2.4.2.4.2

Cancel the common factors.

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Step 1.2.4.2.4.2.1

Factor $4$ out of $16$ .

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

Step 1.2.4.2.4.2.2

Cancel the common factor.

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

Step 1.2.4.2.4.2.3

Rewrite the expression.

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

Step 1.2.5

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

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

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

$h = \frac{3}{4}$

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

Step 3

Find the vertex $(h, k)$ .

$(\frac{3}{4}, - \frac{1}{2})$

Step 4