Precalculus Examples

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

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

Tap for more steps...

Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Add $2$ to both sides of the equation.

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

Step 1.1.2

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

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

Tap for more steps...

Step 1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1.1

Move the negative in front of the fraction.

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

Step 1.1.2.3.1.2

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

Tap for more steps...

Step 1.1.2.3.1.2.1

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

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

Step 1.1.2.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.3.1.2.2.1

Factor $2$ out of $6$ .

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

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

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

Step 1.1.2.3.1.4

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 1.1.2.3.1.4.1

Factor $2$ out of $2$ .

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{2 (1)}{6}$

Step 1.1.2.3.1.4.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.3.1.4.2.1

Factor $2$ out of $6$ .

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{2 \cdot 1}{2 \cdot 3}$

Step 1.1.2.3.1.4.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{2 \cdot 1}{2 \cdot 3}$

Step 1.1.2.3.1.4.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{1}{3}$

Step 1.2

Complete the square for $- \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{1}{3}$ .

Tap for more steps...

Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

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

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

$c = \frac{1}{3}$

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

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

Tap for more steps...

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 $6$ .

Tap for more steps...

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.

Tap for more steps...

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

Tap for more steps...

Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.2.4.2.1.1

Simplify the numerator.

Tap for more steps...

Step 1.2.4.2.1.1.1

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

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

Step 1.2.4.2.1.1.2

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

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

Step 1.2.4.2.1.1.3

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

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

Step 1.2.4.2.1.1.4

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

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

Step 1.2.4.2.1.1.5

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

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

Step 1.2.4.2.1.1.6

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

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

Step 1.2.4.2.1.2

Simplify the denominator.

Tap for more steps...

Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

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

Step 1.2.4.2.1.2.2

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

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

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.2.4.2.1.3.1

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

Tap for more steps...

Step 1.2.4.2.1.3.1.1

Factor $2$ out of $- 4$ .

$e = \frac{1}{3} - \frac{\frac{4}{9}}{\frac{2 (- 2)}{6}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.2.1.3.1.2.1

Factor $2$ out of $6$ .

$e = \frac{1}{3} - \frac{\frac{4}{9}}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = \frac{1}{3} - \frac{\frac{4}{9}}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = \frac{1}{3} - \frac{\frac{4}{9}}{\frac{- 2}{3}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = \frac{1}{3} - \frac{\frac{4}{9}}{- \frac{2}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{3} - (\frac{4}{9} (- \frac{3}{2}))$

Step 1.2.4.2.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.4.2.1.5.1

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

$e = \frac{1}{3} - (\frac{4}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.2

Factor $2$ out of $4$ .

$e = \frac{1}{3} - (\frac{2 (2)}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.3

Cancel the common factor.

$e = \frac{1}{3} - (\frac{2 \cdot 2}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.4

Rewrite the expression.

$e = \frac{1}{3} - (\frac{2}{9} \cdot - 3)$

Step 1.2.4.2.1.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.4.2.1.6.1

Factor $3$ out of $9$ .

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

Step 1.2.4.2.1.6.2

Factor $3$ out of $- 3$ .

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

Step 1.2.4.2.1.6.3

Cancel the common factor.

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

Step 1.2.4.2.1.6.4

Rewrite the expression.

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

Step 1.2.4.2.1.7

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

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

Step 1.2.4.2.1.8

Multiply $2$ by $- 1$ .

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

Step 1.2.4.2.1.9

Move the negative in front of the fraction.

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

Step 1.2.4.2.1.10

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

Tap for more steps...

Step 1.2.4.2.1.10.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.4.2.1.10.2

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

$e = \frac{1}{3} + \frac{2}{3}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{1 + 2}{3}$

Step 1.2.4.2.3

Add $1$ 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

Find the vertex $(h, k)$ .

$(- 2, 1)$

Step 4

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

Tap for more steps...

Step 4.1

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

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

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

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

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

Tap for more steps...

Step 4.3.1.1

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

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

Step 4.3.1.2

Move the negative in front of the fraction.

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

Step 4.3.2

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

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

Step 4.3.3

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

Tap for more steps...

Step 4.3.3.1

Factor $2$ out of $4$ .

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

Step 4.3.3.2

Cancel the common factors.

Tap for more steps...

Step 4.3.3.2.1

Factor $2$ out of $6$ .

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

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

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.5

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

$- \frac{3}{2}$

Step 5

Find the focus.

Tap for more steps...

Step 5.1

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 5.2

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

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

Step 6