Precalculus Examples

$f (x) = 2 x^{2} + x - 6$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $2 x^{2} + x - 6$ .

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Step 1.1.1

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

$a = 2$

$b = 1$

$c = - 6$

Step 1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Step 1.1.3.2

Multiply $2$ by $2$ .

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

Step 1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

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

$e = - 6 - \frac{1^{2}}{4 \cdot 2}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

One to any power is one.

$e = - 6 - \frac{1}{4 \cdot 2}$

Step 1.1.4.2.1.2

Multiply $4$ by $2$ .

$e = - 6 - \frac{1}{8}$

Step 1.1.4.2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = - 6 \cdot \frac{8}{8} - \frac{1}{8}$

Step 1.1.4.2.3

Combine $- 6$ and $\frac{8}{8}$ .

$e = \frac{- 6 \cdot 8}{8} - \frac{1}{8}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 8 - 1}{8}$

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply $- 6$ by $8$ .

$e = \frac{- 48 - 1}{8}$

Step 1.1.4.2.5.2

Subtract $1$ from $- 48$ .

$e = \frac{- 49}{8}$

Step 1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{49}{8}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{1}{4})}^{2} - \frac{49}{8}$ .

$2 {(x + \frac{1}{4})}^{2} - \frac{49}{8}$

Step 1.2

Set $y$ equal to the new right side.

$y = 2 {(x + \frac{1}{4})}^{2} - \frac{49}{8}$

Step 2

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

$a = 2$

$h = - \frac{1}{4}$

$k = - \frac{49}{8}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{1}{4}, - \frac{49}{8})$

Step 4