Precalculus Examples

f (x) = x^{2} - 8 x + 16

$f (x) = x^{2} - 8 x + 16$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

x^{2} - 8 x + 16

$x^{2} - 8 x + 16$ .

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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 = - 8

$b = - 8$

c = 16

$c = 16$

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{- 8}{2 \cdot 1}

$d = \frac{- 8}{2 \cdot 1}$

Step 1.1.3.2

Cancel the common factor of

- 8

$- 8$ and

2

$2$ .

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Step 1.1.3.2.1

Factor

2

$2$ out of

- 8

$- 8$ .

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

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

Step 1.1.3.2.2

Cancel the common factors.

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Step 1.1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

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

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

Step 1.1.3.2.2.2

Cancel the common factor.

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

Step 1.1.3.2.2.3

Rewrite the expression.

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

Step 1.1.3.2.2.4

Divide

$- 4$ by

$1$ .

$d = - 4$

$d = - 4$

$d = - 4$

$d = - 4$

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

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

$e = 16 - \frac{{(- 8)}^{2}}{4 \cdot 1}$

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

$- 8$ to the power of

$2$ .

$e = 16 - \frac{64}{4 \cdot 1}$

Step 1.1.4.2.1.2

Multiply

$4$ by

$1$ .

$e = 16 - \frac{64}{4}$

Step 1.1.4.2.1.3

Divide

$64$ by

$4$ .

$e = 16 - 1 \cdot 16$

Step 1.1.4.2.1.4

Multiply

$- 1$ by

$16$ .

$e = 16 - 16$

$e = 16 - 16$

Step 1.1.4.2.2

Subtract

$16$ from

$16$ .

$e = 0$

$e = 0$

$e = 0$

Step 1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(x - 4)}^{2} + 0$ .

${(x - 4)}^{2} + 0$

${(x - 4)}^{2} + 0$

Step 1.2

Set

$y$ equal to the new right side.

$y = {(x - 4)}^{2} + 0$

$y = {(x - 4)}^{2} + 0$

Step 2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 1$

$h = 4$

$k = 0$

Step 3

Find the vertex

$(h, k)$ .

$(4, 0)$

Step 4

$f (x) = x^{2} - 8 x + 16$

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