Precalculus Examples

$x^{2} + 4 x + 6 y + 10 = 0$

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.

$4 x + 6 y + 10 = - x^{2}$

Step 1.1.1.2

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

$6 y + 10 = - x^{2} - 4 x$

Step 1.1.1.3

Subtract $10$ from both sides of the equation.

$6 y = - x^{2} - 4 x - 10$

Step 1.1.2

Divide each term in $6 y = - x^{2} - 4 x - 10$ by $6$ and simplify.

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Step 1.1.2.1

Divide each term in $6 y = - x^{2} - 4 x - 10$ by $6$ .

$\frac{6 y}{6} = \frac{- x^{2}}{6} + \frac{- 4 x}{6} + \frac{- 10}{6}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $6$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{- x^{2}}{6} + \frac{- 4 x}{6} + \frac{- 10}{6}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{6} + \frac{- 4 x}{6} + \frac{- 10}{6}$

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

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{6} + \frac{- 4 x}{6} + \frac{- 10}{6}$

Step 1.1.2.3.1.2

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

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Step 1.1.2.3.1.2.1

Factor $2$ out of $- 4 x$ .

$y = - \frac{x^{2}}{6} + \frac{2 (- 2 x)}{6} + \frac{- 10}{6}$

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 $6$ .

$y = - \frac{x^{2}}{6} + \frac{2 (- 2 x)}{2 (3)} + \frac{- 10}{6}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{6} + \frac{2 (- 2 x)}{2 \cdot 3} + \frac{- 10}{6}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{6} + \frac{- 2 x}{3} + \frac{- 10}{6}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{- 10}{6}$

Step 1.1.2.3.1.4

Cancel the common factor of $- 10$ and $6$ .

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Step 1.1.2.3.1.4.1

Factor $2$ out of $- 10$ .

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{2 (- 5)}{6}$

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 $2$ out of $6$ .

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{2 \cdot - 5}{2 \cdot 3}$

Step 1.1.2.3.1.4.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{2 \cdot - 5}{2 \cdot 3}$

Step 1.1.2.3.1.4.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{- 5}{3}$

Step 1.1.2.3.1.5

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} - \frac{5}{3}$

Step 1.2

Complete the square for $- \frac{x^{2}}{6} - \frac{2 x}{3} - \frac{5}{3}$ .

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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 $c$ .

$a = - \frac{1}{6}$

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

$c = - \frac{5}{3}$

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

$d = \frac{- \frac{2}{3}}{2 (- \frac{1}{6})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$d = \frac{\frac{2}{3}}{2 (\frac{1}{6})}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{2}{3} \cdot \frac{1}{2 (\frac{1}{6})}$

Step 1.2.3.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.2.3.1

Cancel the common factor.

$d = \frac{2}{3} \cdot \frac{1}{2 (\frac{1}{6})}$

Step 1.2.3.2.3.2

Rewrite the expression.

$d = \frac{1}{3} \cdot \frac{1}{\frac{1}{6}}$

Step 1.2.3.2.4

Multiply $\frac{1}{3}$ by $\frac{1}{\frac{1}{6}}$ .

$d = \frac{1}{3 (\frac{1}{6})}$

Step 1.2.3.2.5

Combine $3$ and $\frac{1}{6}$ .

$d = \frac{1}{\frac{3}{6}}$

Step 1.2.3.2.6

Cancel the common factor of $3$ and $6$ .

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Step 1.2.3.2.6.1

Factor $3$ out of $3$ .

$d = \frac{1}{\frac{3 (1)}{6}}$

Step 1.2.3.2.6.2

Cancel the common factors.

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Step 1.2.3.2.6.2.1

Factor $3$ out of $6$ .

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

Step 1.2.3.2.6.2.2

Cancel the common factor.

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

Step 1.2.3.2.6.2.3

Rewrite the expression.

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

Step 1.2.3.2.7

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

Step 1.2.3.2.8

Multiply $2$ by $1$ .

$d = 2$

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

$e = - \frac{5}{3} - \frac{{(- \frac{2}{3})}^{2}}{4 (- \frac{1}{6})}$

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 $- \frac{2}{3}$ .

$e = - \frac{5}{3} - \frac{{(- 1)}^{2} {(\frac{2}{3})}^{2}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.2

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

$e = - \frac{5}{3} - \frac{1 {(\frac{2}{3})}^{2}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{2}{3}$ .

$e = - \frac{5}{3} - \frac{1 \frac{2^{2}}{3^{2}}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.4

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

$e = - \frac{5}{3} - \frac{1 \frac{4}{3^{2}}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.5

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

$e = - \frac{5}{3} - \frac{1 (\frac{4}{9})}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.6

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

$e = - \frac{5}{3} - \frac{\frac{4}{9}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{5}{3} - \frac{\frac{4}{9}}{- 4 (\frac{1}{6})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{6}$ .

$e = - \frac{5}{3} - \frac{\frac{4}{9}}{\frac{- 4}{6}}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.3.1

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

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Step 1.2.4.2.1.3.1.1

Factor $2$ out of $- 4$ .

$e = - \frac{5}{3} - \frac{\frac{4}{9}}{\frac{2 (- 2)}{6}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

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Step 1.2.4.2.1.3.1.2.1

Factor $2$ out of $6$ .

$e = - \frac{5}{3} - \frac{\frac{4}{9}}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{5}{3} - \frac{\frac{4}{9}}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{5}{3} - \frac{\frac{4}{9}}{\frac{- 2}{3}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{5}{3} - \frac{\frac{4}{9}}{- \frac{2}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{5}{3} - (\frac{4}{9} (- \frac{3}{2}))$

Step 1.2.4.2.1.5

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.5.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$e = - \frac{5}{3} - (\frac{4}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.2

Factor $2$ out of $4$ .

$e = - \frac{5}{3} - (\frac{2 (2)}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.3

Cancel the common factor.

$e = - \frac{5}{3} - (\frac{2 \cdot 2}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.4

Rewrite the expression.

$e = - \frac{5}{3} - (\frac{2}{9} \cdot - 3)$

Step 1.2.4.2.1.6

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.6.1

Factor $3$ out of $9$ .

$e = - \frac{5}{3} - (\frac{2}{3 (3)} \cdot - 3)$

Step 1.2.4.2.1.6.2

Factor $3$ out of $- 3$ .

$e = - \frac{5}{3} - (\frac{2}{3 \cdot 3} \cdot (3 \cdot - 1))$

Step 1.2.4.2.1.6.3

Cancel the common factor.

$e = - \frac{5}{3} - (\frac{2}{3 \cdot 3} \cdot (3 \cdot - 1))$

Step 1.2.4.2.1.6.4

Rewrite the expression.

$e = - \frac{5}{3} - (\frac{2}{3} \cdot - 1)$

Step 1.2.4.2.1.7

Combine $\frac{2}{3}$ and $- 1$ .

$e = - \frac{5}{3} - \frac{2 \cdot - 1}{3}$

Step 1.2.4.2.1.8

Multiply $2$ by $- 1$ .

$e = - \frac{5}{3} - \frac{- 2}{3}$

Step 1.2.4.2.1.9

Move the negative in front of the fraction.

$e = - \frac{5}{3} - - \frac{2}{3}$

Step 1.2.4.2.1.10

Multiply $- - \frac{2}{3}$ .

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Step 1.2.4.2.1.10.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{5}{3} + 1 (\frac{2}{3})$

Step 1.2.4.2.1.10.2

Multiply $\frac{2}{3}$ by $1$ .

$e = - \frac{5}{3} + \frac{2}{3}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 5 + 2}{3}$

Step 1.2.4.2.3

Add $- 5$ and $2$ .

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

Step 1.2.4.2.4

Divide $- 3$ by $3$ .

$e = - 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{6} {(x + 2)}^{2} - 1$ .

$- \frac{1}{6} {(x + 2)}^{2} - 1$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{6} \cdot {(x + 2)}^{2} - 1$

Step 2

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

$a = - \frac{1}{6}$

$h = - 2$

$k = - 1$

Step 3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(- 2, - 1)$

Step 5

Find $p$ , the distance from the vertex to the focus.

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{6})}$

Step 5.3

Simplify.

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Step 5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{6})}$

Step 5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{6})}$

Step 5.3.2

Combine $4$ and $\frac{1}{6}$ .

$- \frac{1}{\frac{4}{6}}$

Step 5.3.3

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

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Step 5.3.3.1

Factor $2$ out of $4$ .

$- \frac{1}{\frac{2 (2)}{6}}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $2$ out of $6$ .

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 5.3.3.2.2

Cancel the common factor.

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 5.3.3.2.3

Rewrite the expression.

$- \frac{1}{\frac{2}{3}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.5

Multiply $\frac{3}{2}$ by $1$ .

$- \frac{3}{2}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- 2, - \frac{5}{2})$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = - 2$

Step 8

Find the directrix.

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Step 8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 8.2

Substitute the known values of $p$ and $k$ into the formula and simplify.

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

Step 9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: $(- 2, - 1)$

Focus: $(- 2, - \frac{5}{2})$

Axis of Symmetry: $x = - 2$

Directrix: $y = \frac{1}{2}$

Step 10