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Examples

f (x) = 2 x^{2} + 3 x - 4

$f (x) = 2 x^{2} + 3 x - 4$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

$2 x^{2} + 3 x - 4$ .

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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

$c$ .

$a = 2$

$b = 3$

$c = - 4$

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

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

$d = \frac{3}{2 \cdot 2}$

Step 1.1.3.2

Multiply

$2$ by

$2$ .

$d = \frac{3}{4}$

$d = \frac{3}{4}$

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

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

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

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

Raise

$3$ to the power of

$2$ .

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

Step 1.1.4.2.1.2

Multiply

$4$ by

$2$ .

$e = - 4 - \frac{9}{8}$

$e = - 4 - \frac{9}{8}$

Step 1.1.4.2.2

To write

$- 4$ as a fraction with a common denominator, multiply by

$\frac{8}{8}$ .

$e = - 4 \cdot \frac{8}{8} - \frac{9}{8}$

Step 1.1.4.2.3

Combine

$- 4$ and

$\frac{8}{8}$ .

$e = \frac{- 4 \cdot 8}{8} - \frac{9}{8}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 4 \cdot 8 - 9}{8}$

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply

$- 4$ by

$8$ .

$e = \frac{- 32 - 9}{8}$

Step 1.1.4.2.5.2

Subtract

$9$ from

$- 32$ .

$e = \frac{- 41}{8}$

$e = \frac{- 41}{8}$

Step 1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{41}{8}$

$e = - \frac{41}{8}$

$e = - \frac{41}{8}$

Step 1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$2 {(x + \frac{3}{4})}^{2} - \frac{41}{8}$ .

$2 {(x + \frac{3}{4})}^{2} - \frac{41}{8}$

$2 {(x + \frac{3}{4})}^{2} - \frac{41}{8}$

Step 1.2

Set

$y$ equal to the new right side.

$y = 2 {(x + \frac{3}{4})}^{2} - \frac{41}{8}$

$y = 2 {(x + \frac{3}{4})}^{2} - \frac{41}{8}$

Step 2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 2$

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

$k = - \frac{41}{8}$

Step 3

Find the vertex

$(h, k)$ .

$(- \frac{3}{4}, - \frac{41}{8})$

Step 4

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$