Precalculus Examples

$x^{2} + 8 x = 4 y - 12$

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

Rewrite the equation as $4 y - 12 = x^{2} + 8 x$ .

$4 y - 12 = x^{2} + 8 x$

Step 1.1.2

Add $12$ to both sides of the equation.

$4 y = x^{2} + 8 x + 12$

Step 1.1.3

Divide each term in $4 y = x^{2} + 8 x + 12$ by $4$ and simplify.

Tap for more steps...

Step 1.1.3.1

Divide each term in $4 y = x^{2} + 8 x + 12$ by $4$ .

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{8 x}{4} + \frac{12}{4}$

Step 1.1.3.2

Simplify the left side.

Tap for more steps...

Step 1.1.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{8 x}{4} + \frac{12}{4}$

Step 1.1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{4} + \frac{8 x}{4} + \frac{12}{4}$

Step 1.1.3.3

Simplify the right side.

Tap for more steps...

Step 1.1.3.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.3.1.1

Cancel the common factor of $8$ and $4$ .

Tap for more steps...

Step 1.1.3.3.1.1.1

Factor $4$ out of $8 x$ .

$y = \frac{x^{2}}{4} + \frac{4 (2 x)}{4} + \frac{12}{4}$

Step 1.1.3.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.3.3.1.1.2.1

Factor $4$ out of $4$ .

$y = \frac{x^{2}}{4} + \frac{4 (2 x)}{4 (1)} + \frac{12}{4}$

Step 1.1.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{2}}{4} + \frac{4 (2 x)}{4 \cdot 1} + \frac{12}{4}$

Step 1.1.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{4} + \frac{2 x}{1} + \frac{12}{4}$

Step 1.1.3.3.1.1.2.4

Divide $2 x$ by $1$ .

$y = \frac{x^{2}}{4} + 2 x + \frac{12}{4}$

Step 1.1.3.3.1.2

Divide $12$ by $4$ .

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

Step 1.2

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

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 $c$ .

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

$b = 2$

$c = 3$

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

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

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

Rewrite the expression.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 4$

Step 1.2.3.2.3

Multiply $4$ by $1$ .

$d = 4$

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

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

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

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

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

Step 1.2.4.2.1.2

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

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

Step 1.2.4.2.1.3

Divide $4$ by $4$ .

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

Step 1.2.4.2.1.4

Divide $4$ by $1$ .

$e = 3 - 1 \cdot 4$

Step 1.2.4.2.1.5

Multiply $- 1$ by $4$ .

$e = 3 - 4$

Step 1.2.4.2.2

Subtract $4$ from $3$ .

$e = - 1$

Step 1.2.5

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

$\frac{1}{4} {(x + 4)}^{2} - 1$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = - 4$

$k = - 1$

Step 3

Find the vertex $(h, k)$ .

$(- 4, - 1)$

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.

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

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

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

$\frac{1}{\frac{4}{4}}$

Step 4.3.2

Simplify by dividing numbers.

Tap for more steps...

Step 4.3.2.1

Divide $4$ by $4$ .

$\frac{1}{1}$

Step 4.3.2.2

Divide $1$ by $1$ .

$1$

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.

$(- 4, 0)$

Step 6