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Precalculus Examples

f (x) = 8 x - 4 + 2 x^{2}

$f (x) = 8 x - 4 + 2 x^{2}$

Step 1

Write

f (x) = 8 x - 4 + 2 x^{2}

$f (x) = 8 x - 4 + 2 x^{2}$ as an equation.

y = 8 x - 4 + 2 x^{2}

$y = 8 x - 4 + 2 x^{2}$

Step 2

Simplify

8 x - 4 + 2 x^{2}

$8 x - 4 + 2 x^{2}$ .

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Step 2.1

Move

- 4

$- 4$ .

y = 8 x + 2 x^{2} - 4

$y = 8 x + 2 x^{2} - 4$

Step 2.2

Reorder

8 x

$8 x$ and

2 x^{2}

$2 x^{2}$ .

y = 2 x^{2} + 8 x - 4

$y = 2 x^{2} + 8 x - 4$

y = 2 x^{2} + 8 x - 4

$y = 2 x^{2} + 8 x - 4$

Step 3

Complete the square for

2 x^{2} + 8 x - 4

$2 x^{2} + 8 x - 4$ .

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Step 3.1

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 = 2

$a = 2$

b = 8

$b = 8$

c = - 4

$c = - 4$

Step 3.2

Consider the vertex form of a parabola.

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

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

Step 3.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 3.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{8}{2 \cdot 2}

$d = \frac{8}{2 \cdot 2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of

8

$8$ and

2

$2$ .

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Step 3.3.2.1.1

Factor

2

$2$ out of

8

$8$ .

d = \frac{2 \cdot 4}{2 \cdot 2}

$d = \frac{2 \cdot 4}{2 \cdot 2}$

Step 3.3.2.1.2

Cancel the common factors.

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Step 3.3.2.1.2.1

Factor

2

$2$ out of

2 \cdot 2

$2 \cdot 2$ .

d = \frac{2 \cdot 4}{2 (2)}

$d = \frac{2 \cdot 4}{2 (2)}$

Step 3.3.2.1.2.2

Cancel the common factor.

d = \frac{2 \cdot 4}{2 \cdot 2}

$d = \frac{2 \cdot 4}{2 \cdot 2}$

Step 3.3.2.1.2.3

Rewrite the expression.

d = \frac{4}{2}

$d = \frac{4}{2}$

d = \frac{4}{2}

$d = \frac{4}{2}$

d = \frac{4}{2}

$d = \frac{4}{2}$

Step 3.3.2.2

Cancel the common factor of

4

$4$ and

2

$2$ .

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Step 3.3.2.2.1

Factor

2

$2$ out of

4

$4$ .

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

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

Step 3.3.2.2.2

Cancel the common factors.

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Step 3.3.2.2.2.1

Factor

2

$2$ out of

2

$2$ .

d = \frac{2 \cdot 2}{2 (1)}

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

Step 3.3.2.2.2.2

Cancel the common factor.

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

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

Step 3.3.2.2.2.3

Rewrite the expression.

d = \frac{2}{1}

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

Step 3.3.2.2.2.4

Divide

2

$2$ by

1

$1$ .

d = 2

$d = 2$

d = 2

$d = 2$

d = 2

$d = 2$

d = 2

$d = 2$

d = 2

$d = 2$

Step 3.4

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 3.4.1

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 = - 4 - \frac{8^{2}}{4 \cdot 2}

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Raise

8

$8$ to the power of

2

$2$ .

e = - 4 - \frac{64}{4 \cdot 2}

$e = - 4 - \frac{64}{4 \cdot 2}$

Step 3.4.2.1.2

Multiply

4

$4$ by

2

$2$ .

e = - 4 - \frac{64}{8}

$e = - 4 - \frac{64}{8}$

Step 3.4.2.1.3

Divide

64

$64$ by

8

$8$ .

e = - 4 - 1 \cdot 8

$e = - 4 - 1 \cdot 8$

Step 3.4.2.1.4

Multiply

- 1

$- 1$ by

8

$8$ .

e = - 4 - 8

$e = - 4 - 8$

e = - 4 - 8

$e = - 4 - 8$

Step 3.4.2.2

Subtract

8

$8$ from

- 4

$- 4$ .

e = - 12

$e = - 12$

e = - 12

$e = - 12$

e = - 12

$e = - 12$

Step 3.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

2 {(x + 2)}^{2} - 12

$2 {(x + 2)}^{2} - 12$ .

2 {(x + 2)}^{2} - 12

$2 {(x + 2)}^{2} - 12$

2 {(x + 2)}^{2} - 12

$2 {(x + 2)}^{2} - 12$

Step 4

Set

y

$y$ equal to the new right side.

y = 2 {(x + 2)}^{2} - 12

$y = 2 {(x + 2)}^{2} - 12$

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

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