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Precalculus Examples

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

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

Step 1

Write

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

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

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

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

Step 2

Complete the square for

x^{2} - 4 x + 2

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

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Step 2.1

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 = - 4

$b = - 4$

c = 2

$c = 2$

Step 2.2

Consider the vertex form of a parabola.

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

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

Step 2.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 2.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{- 4}{2 \cdot 1}

$d = \frac{- 4}{2 \cdot 1}$

Step 2.3.2

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

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Step 2.3.2.1

Factor

2

$2$ out of

- 4

$- 4$ .

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

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

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

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide

$- 2$ by

$1$ .

$d = - 2$

$d = - 2$

$d = - 2$

$d = - 2$

Step 2.4

Find the value of

$e$ using the formula

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

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Step 2.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Cancel the common factor of

${(- 4)}^{2}$ and

$4$ .

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Step 2.4.2.1.1.1

Rewrite

$- 4$ as

$- 1 (4)$ .

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

Step 2.4.2.1.1.2

Apply the product rule to

$- 1 (4)$ .

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

Step 2.4.2.1.1.3

Raise

$- 1$ to the power of

$2$ .

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

Step 2.4.2.1.1.4

Multiply

$4^{2}$ by

$1$ .

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

Step 2.4.2.1.1.5

Factor

$4$ out of

$4^{2}$ .

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

Step 2.4.2.1.1.6

Cancel the common factors.

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Step 2.4.2.1.1.6.1

Factor

$4$ out of

$4 \cdot 1$ .

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

Step 2.4.2.1.1.6.2

Cancel the common factor.

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

Step 2.4.2.1.1.6.3

Rewrite the expression.

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

Step 2.4.2.1.1.6.4

Divide

$4$ by

$1$ .

$e = 2 - 1 \cdot 4$

$e = 2 - 1 \cdot 4$

$e = 2 - 1 \cdot 4$

Step 2.4.2.1.2

Multiply

$- 1$ by

$4$ .

$e = 2 - 4$

$e = 2 - 4$

Step 2.4.2.2

Subtract

$4$ from

$2$ .

$e = - 2$

$e = - 2$

$e = - 2$

Step 2.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

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

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

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

Step 3

Set

$y$ equal to the new right side.

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

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