Precalculus Examples

$2 y^{2} - 15 x - 4 y - 28 = 0$

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

$- 15 x - 4 y - 28 = - 2 y^{2}$

Step 1.1.1.2

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

$- 15 x - 28 = - 2 y^{2} + 4 y$

Step 1.1.1.3

Add $28$ to both sides of the equation.

$- 15 x = - 2 y^{2} + 4 y + 28$

Step 1.1.2

Divide each term in $- 15 x = - 2 y^{2} + 4 y + 28$ by $- 15$ and simplify.

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Step 1.1.2.1

Divide each term in $- 15 x = - 2 y^{2} + 4 y + 28$ by $- 15$ .

$\frac{- 15 x}{- 15} = \frac{- 2 y^{2}}{- 15} + \frac{4 y}{- 15} + \frac{28}{- 15}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 15$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 15 x}{- 15} = \frac{- 2 y^{2}}{- 15} + \frac{4 y}{- 15} + \frac{28}{- 15}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 y^{2}}{- 15} + \frac{4 y}{- 15} + \frac{28}{- 15}$

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.

$x = \frac{2 y^{2}}{15} + \frac{4 y}{- 15} + \frac{28}{- 15}$

Step 1.1.2.3.1.2

Move the negative in front of the fraction.

$x = \frac{2 y^{2}}{15} - \frac{4 y}{15} + \frac{28}{- 15}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$x = \frac{2 y^{2}}{15} - \frac{4 y}{15} - \frac{28}{15}$

Step 1.2

Complete the square for $\frac{2 y^{2}}{15} - \frac{4 y}{15} - \frac{28}{15}$ .

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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 $c$ .

$a = \frac{2}{15}$

$b = - \frac{4}{15}$

$c = - \frac{28}{15}$

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

$d = \frac{- \frac{4}{15}}{2 (\frac{2}{15})}$

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.

$d = - \frac{4}{15} \cdot \frac{1}{2 (\frac{2}{15})}$

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Move the leading negative in $- \frac{4}{15}$ into the numerator.

$d = \frac{- 4}{15} \cdot \frac{1}{2 (\frac{2}{15})}$

Step 1.2.3.2.2.2

Factor $2$ out of $- 4$ .

$d = \frac{2 (- 2)}{15} \cdot \frac{1}{2 (\frac{2}{15})}$

Step 1.2.3.2.2.3

Cancel the common factor.

$d = \frac{2 \cdot - 2}{15} \cdot \frac{1}{2 (\frac{2}{15})}$

Step 1.2.3.2.2.4

Rewrite the expression.

$d = \frac{- 2}{15} \cdot \frac{1}{\frac{2}{15}}$

Step 1.2.3.2.3

Multiply $\frac{- 2}{15}$ by $\frac{1}{\frac{2}{15}}$ .

$d = \frac{- 2}{15 (\frac{2}{15})}$

Step 1.2.3.2.4

Combine $15$ and $\frac{2}{15}$ .

$d = \frac{- 2}{\frac{15 \cdot 2}{15}}$

Step 1.2.3.2.5

Multiply $15$ by $2$ .

$d = \frac{- 2}{\frac{30}{15}}$

Step 1.2.3.2.6

Cancel the common factor of $30$ and $15$ .

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Step 1.2.3.2.6.1

Factor $15$ out of $30$ .

$d = \frac{- 2}{\frac{15 \cdot 2}{15}}$

Step 1.2.3.2.6.2

Cancel the common factors.

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Step 1.2.3.2.6.2.1

Factor $15$ out of $15$ .

$d = \frac{- 2}{\frac{15 \cdot 2}{15 (1)}}$

Step 1.2.3.2.6.2.2

Cancel the common factor.

$d = \frac{- 2}{\frac{15 \cdot 2}{15 \cdot 1}}$

Step 1.2.3.2.6.2.3

Rewrite the expression.

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

Step 1.2.3.2.6.2.4

Divide $2$ by $1$ .

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

Step 1.2.3.2.7

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

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Step 1.2.3.2.7.1

Factor $2$ out of $- 2$ .

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

Step 1.2.3.2.7.2

Cancel the common factors.

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Step 1.2.3.2.7.2.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.7.2.2

Cancel the common factor.

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

Step 1.2.3.2.7.2.3

Rewrite the expression.

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

Step 1.2.3.2.7.2.4

Divide $- 1$ by $1$ .

$d = - 1$

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

$e = - \frac{28}{15} - \frac{{(- \frac{4}{15})}^{2}}{4 (\frac{2}{15})}$

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 $- \frac{4}{15}$ .

$e = - \frac{28}{15} - \frac{{(- 1)}^{2} {(\frac{4}{15})}^{2}}{4 (\frac{2}{15})}$

Step 1.2.4.2.1.1.2

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

$e = - \frac{28}{15} - \frac{1 {(\frac{4}{15})}^{2}}{4 (\frac{2}{15})}$

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{4}{15}$ .

$e = - \frac{28}{15} - \frac{1 \frac{4^{2}}{15^{2}}}{4 (\frac{2}{15})}$

Step 1.2.4.2.1.1.4

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

$e = - \frac{28}{15} - \frac{1 \frac{16}{15^{2}}}{4 (\frac{2}{15})}$

Step 1.2.4.2.1.1.5

Raise $15$ to the power of $2$ .

$e = - \frac{28}{15} - \frac{1 (\frac{16}{225})}{4 (\frac{2}{15})}$

Step 1.2.4.2.1.1.6

Multiply $\frac{16}{225}$ by $1$ .

$e = - \frac{28}{15} - \frac{\frac{16}{225}}{4 (\frac{2}{15})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{2}{15}$ .

$e = - \frac{28}{15} - \frac{\frac{16}{225}}{\frac{4 \cdot 2}{15}}$

Step 1.2.4.2.1.3

Multiply $4$ by $2$ .

$e = - \frac{28}{15} - \frac{\frac{16}{225}}{\frac{8}{15}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{28}{15} - (\frac{16}{225} \cdot \frac{15}{8})$

Step 1.2.4.2.1.5

Cancel the common factor of $8$ .

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Step 1.2.4.2.1.5.1

Factor $8$ out of $16$ .

$e = - \frac{28}{15} - (\frac{8 (2)}{225} \cdot \frac{15}{8})$

Step 1.2.4.2.1.5.2

Cancel the common factor.

$e = - \frac{28}{15} - (\frac{8 \cdot 2}{225} \cdot \frac{15}{8})$

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = - \frac{28}{15} - (\frac{2}{225} \cdot 15)$

Step 1.2.4.2.1.6

Cancel the common factor of $15$ .

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Step 1.2.4.2.1.6.1

Factor $15$ out of $225$ .

$e = - \frac{28}{15} - (\frac{2}{15 (15)} \cdot 15)$

Step 1.2.4.2.1.6.2

Cancel the common factor.

$e = - \frac{28}{15} - (\frac{2}{15 \cdot 15} \cdot 15)$

Step 1.2.4.2.1.6.3

Rewrite the expression.

$e = - \frac{28}{15} - \frac{2}{15}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 28 - 2}{15}$

Step 1.2.4.2.3

Subtract $2$ from $- 28$ .

$e = \frac{- 30}{15}$

Step 1.2.4.2.4

Divide $- 30$ by $15$ .

$e = - 2$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{2}{15} {(y - 1)}^{2} - 2$ .

$\frac{2}{15} {(y - 1)}^{2} - 2$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{2}{15} \cdot {(y - 1)}^{2} - 2$

Step 2

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

$a = \frac{2}{15}$

$h = - 2$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 1)$

Step 4