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Algebra Examples

f (x) = 2 x^{2} + 16 x - 1

$f (x) = 2 x^{2} + 16 x - 1$

Step 1

Write

f (x) = 2 x^{2} + 16 x - 1

$f (x) = 2 x^{2} + 16 x - 1$ as an equation.

y = 2 x^{2} + 16 x - 1

$y = 2 x^{2} + 16 x - 1$

Step 2

Complete the square for

2 x^{2} + 16 x - 1

$2 x^{2} + 16 x - 1$ .

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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 = 2

$a = 2$

b = 16

$b = 16$

c = - 1

$c = - 1$

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{16}{2 \cdot 2}

$d = \frac{16}{2 \cdot 2}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of

16

$16$ and

2

$2$ .

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Step 2.3.2.1.1

Factor

2

$2$ out of

16

$16$ .

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

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

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Factor

2

$2$ out of

2 \cdot 2

$2 \cdot 2$ .

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

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

Step 2.3.2.1.2.2

Cancel the common factor.

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

Step 2.3.2.1.2.3

Rewrite the expression.

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

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

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

Step 2.3.2.2

Cancel the common factor of

$8$ and

$2$ .

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Step 2.3.2.2.1

Factor

$2$ out of

$8$ .

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

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor

$2$ out of

$2$ .

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

Step 2.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide

$4$ by

$1$ .

$d = 4$

$d = 4$

$d = 4$

$d = 4$

$d = 4$

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

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

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

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

Raise

$16$ to the power of

$2$ .

$e = - 1 - \frac{256}{4 \cdot 2}$

Step 2.4.2.1.2

Multiply

$4$ by

$2$ .

$e = - 1 - \frac{256}{8}$

Step 2.4.2.1.3

Divide

$256$ by

$8$ .

$e = - 1 - 1 \cdot 32$

Step 2.4.2.1.4

Multiply

$- 1$ by

$32$ .

$e = - 1 - 32$

$e = - 1 - 32$

Step 2.4.2.2

Subtract

$32$ from

$- 1$ .

$e = - 33$

$e = - 33$

$e = - 33$

Step 2.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$2 {(x + 4)}^{2} - 33$ .

$2 {(x + 4)}^{2} - 33$

$2 {(x + 4)}^{2} - 33$

Step 3

Set

$y$ equal to the new right side.

$y = 2 {(x + 4)}^{2} - 33$

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