Precalculus Examples

$x + 8 = \frac{1}{5} y^{2}$

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

Simplify $\frac{1}{5} y^{2}$ .

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Step 1.1.1.1

Rewrite.

$x + 8 = 0 + 0 + \frac{1}{5} y^{2}$

Step 1.1.1.2

Simplify by adding zeros.

$x + 8 = \frac{1}{5} y^{2}$

Step 1.1.1.3

Combine $\frac{1}{5}$ and $y^{2}$ .

$x + 8 = \frac{y^{2}}{5}$

Step 1.1.2

Subtract $8$ from both sides of the equation.

$x = \frac{y^{2}}{5} - 8$

Step 1.2

Complete the square for $\frac{y^{2}}{5} - 8$ .

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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 $c$ .

$a = \frac{1}{5}$

$b = 0$

$c = - 8$

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

$d = \frac{0}{2 (\frac{1}{5})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{1}{5})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{1}{5})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{\frac{1}{5}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 \cdot 5$

Step 1.2.3.2.3

Multiply $0$ by $5$ .

$d = 0$

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

$e = - 8 - \frac{0^{2}}{4 (\frac{1}{5})}$

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

Raising $0$ to any positive power yields $0$ .

$e = - 8 - \frac{0}{4 (\frac{1}{5})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{5}$ .

$e = - 8 - \frac{0}{\frac{4}{5}}$

Step 1.2.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = - 8 - (0 (\frac{5}{4}))$

Step 1.2.4.2.1.4

Multiply $0$ by $\frac{5}{4}$ .

$e = - 8 - 0$

Step 1.2.4.2.1.5

Multiply $- 1$ by $0$ .

$e = - 8 + 0$

Step 1.2.4.2.2

Add $- 8$ and $0$ .

$e = - 8$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{5} {(y + 0)}^{2} - 8$ .

$\frac{1}{5} {(y + 0)}^{2} - 8$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{5} \cdot {(y + 0)}^{2} - 8$

Step 2

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

$a = \frac{1}{5}$

$h = - 8$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(- 8, 0)$

Step 4