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Examples

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

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

x^{2} + 3 x - 3

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

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Step 1.1.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 1

$a = 1$

b = 3

$b = 3$

c = - 3

$c = - 3$

Step 1.1.2

Consider the vertex form of a parabola.

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

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

Step 1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 1.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{3}{2 \cdot 1}

$d = \frac{3}{2 \cdot 1}$

Step 1.1.3.2

Multiply

2

$2$ by

1

$1$ .

d = \frac{3}{2}

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

d = \frac{3}{2}

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

Step 1.1.4

Find the value of

e

$e$ using the formula

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

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

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Step 1.1.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

e = - 3 - \frac{3^{2}}{4 \cdot 1}

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

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

$3$ to the power of

2

$2$ .

e = - 3 - \frac{9}{4 \cdot 1}

$e = - 3 - \frac{9}{4 \cdot 1}$

Step 1.1.4.2.1.2

Multiply

4

$4$ by

1

$1$ .

e = - 3 - \frac{9}{4}

$e = - 3 - \frac{9}{4}$

e = - 3 - \frac{9}{4}

$e = - 3 - \frac{9}{4}$

Step 1.1.4.2.2

To write

- 3

$- 3$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

e = - 3 \cdot \frac{4}{4} - \frac{9}{4}

$e = - 3 \cdot \frac{4}{4} - \frac{9}{4}$

Step 1.1.4.2.3

Combine

- 3

$- 3$ and

\frac{4}{4}

$\frac{4}{4}$ .

e = \frac{- 3 \cdot 4}{4} - \frac{9}{4}

$e = \frac{- 3 \cdot 4}{4} - \frac{9}{4}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

e = \frac{- 3 \cdot 4 - 9}{4}

$e = \frac{- 3 \cdot 4 - 9}{4}$

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply

- 3

$- 3$ by

4

$4$ .

e = \frac{- 12 - 9}{4}

$e = \frac{- 12 - 9}{4}$

Step 1.1.4.2.5.2

Subtract

9

$9$ from

- 12

$- 12$ .

e = \frac{- 21}{4}

$e = \frac{- 21}{4}$

e = \frac{- 21}{4}

$e = \frac{- 21}{4}$

Step 1.1.4.2.6

Move the negative in front of the fraction.

e = - \frac{21}{4}

$e = - \frac{21}{4}$

e = - \frac{21}{4}

$e = - \frac{21}{4}$

e = - \frac{21}{4}

$e = - \frac{21}{4}$

Step 1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

{(x + \frac{3}{2})}^{2} - \frac{21}{4}

${(x + \frac{3}{2})}^{2} - \frac{21}{4}$ .

{(x + \frac{3}{2})}^{2} - \frac{21}{4}

${(x + \frac{3}{2})}^{2} - \frac{21}{4}$

{(x + \frac{3}{2})}^{2} - \frac{21}{4}

${(x + \frac{3}{2})}^{2} - \frac{21}{4}$

Step 1.2

Set

y

$y$ equal to the new right side.

y = {(x + \frac{3}{2})}^{2} - \frac{21}{4}

$y = {(x + \frac{3}{2})}^{2} - \frac{21}{4}$

y = {(x + \frac{3}{2})}^{2} - \frac{21}{4}

$y = {(x + \frac{3}{2})}^{2} - \frac{21}{4}$

Step 2

Use the vertex form,

y = a {(x - h)}^{2} + k

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

a

$a$ ,

h

$h$ , and

k

$k$ .

a = 1

$a = 1$

h = - \frac{3}{2}

$h = - \frac{3}{2}$

k = - \frac{21}{4}

$k = - \frac{21}{4}$

Step 3

Find the vertex

(h, k)

$(h, k)$ .

(- \frac{3}{2}, - \frac{21}{4})

$(- \frac{3}{2}, - \frac{21}{4})$

Step 4

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⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

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