Precalculus Examples

$2 y^{2} = x - 8$

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

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

$x - 8 = 2 y^{2}$

Step 1.1.2

Add $8$ to both sides of the equation.

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

Step 1.2

Complete the square for $2 y^{2} + 8$ .

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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 $c$ .

$a = 2$

$b = 0$

$c = 8$

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

$d = \frac{0}{2 \cdot 2}$

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 $0$ .

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

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 2$ .

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 2}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{0}{2}$

Step 1.2.3.2.2

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

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Step 1.2.3.2.2.1

Factor $2$ out of $0$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $0$ by $1$ .

$d = 0$

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

$e = 8 - \frac{0^{2}}{4 \cdot 2}$

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 $0$ .

$e = 8 - \frac{0}{4 \cdot 2}$

Step 1.2.4.2.1.2

Multiply $4$ by $2$ .

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

Step 1.2.4.2.1.3

Divide $0$ by $8$ .

$e = 8 - 0$

Step 1.2.4.2.1.4

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 $2 {(y + 0)}^{2} + 8$ .

$2 {(y + 0)}^{2} + 8$

Step 1.3

Set $x$ equal to the new right side.

$x = 2 {(y + 0)}^{2} + 8$

Step 2

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

$a = 2$

$h = 8$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(8, 0)$

Step 4