Algebra Examples

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

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

Step 1

Write

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

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

y = x^{2} - x

$y = x^{2} - x$

Step 2

Complete the square for

x^{2} - x

$x^{2} - x$ .

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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 = - 1

$b = - 1$

c = 0

$c = 0$

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{- 1}{2 \cdot 1}

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of

- 1

$- 1$ and

1

$1$ .

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Step 2.3.2.1.1

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

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

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

Step 2.3.2.1.2

Cancel the common factor.

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

Step 2.3.2.1.3

Rewrite the expression.

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

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

Step 2.3.2.2

Move the negative in front of the fraction.

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

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

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

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

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

$e = 0 - \frac{{(- 1)}^{2}}{4 \cdot 1}$

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

Raise

$- 1$ to the power of

$2$ .

$e = 0 - \frac{1}{4 \cdot 1}$

Step 2.4.2.1.2

Multiply

$4$ by

$1$ .

$e = 0 - \frac{1}{4}$

$e = 0 - \frac{1}{4}$

Step 2.4.2.2

Subtract

$\frac{1}{4}$ from

$0$ .

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

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

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

Step 2.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(x - \frac{1}{2})}^{2} - \frac{1}{4}$ .

${(x - \frac{1}{2})}^{2} - \frac{1}{4}$

${(x - \frac{1}{2})}^{2} - \frac{1}{4}$

Step 3

Set

$y$ equal to the new right side.

$y = {(x - \frac{1}{2})}^{2} - \frac{1}{4}$

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