Algebra Examples

${(\frac{i - 1}{2})}^{2} + \frac{i - 1}{2}$

Step 1

Simplify each term.

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Step 1.1

Split the fraction $\frac{i - 1}{2}$ into two fractions.

${(\frac{i}{2} + \frac{- 1}{2})}^{2} + \frac{i - 1}{2}$

Step 1.2

Move the negative in front of the fraction.

${(\frac{i}{2} - \frac{1}{2})}^{2} + \frac{i - 1}{2}$

Step 1.3

Rewrite ${(\frac{i}{2} - \frac{1}{2})}^{2}$ as $(\frac{i}{2} - \frac{1}{2}) (\frac{i}{2} - \frac{1}{2})$ .

$(\frac{i}{2} - \frac{1}{2}) (\frac{i}{2} - \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.4

Expand $(\frac{i}{2} - \frac{1}{2}) (\frac{i}{2} - \frac{1}{2})$ using the FOIL Method.

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Step 1.4.1

Apply the distributive property.

$\frac{i}{2} (\frac{i}{2} - \frac{1}{2}) - \frac{1}{2} (\frac{i}{2} - \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.4.2

Apply the distributive property.

$\frac{i}{2} \cdot \frac{i}{2} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} (\frac{i}{2} - \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.4.3

Apply the distributive property.

$\frac{i}{2} \cdot \frac{i}{2} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5

Simplify and combine like terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $\frac{i}{2} \cdot \frac{i}{2}$ .

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Step 1.5.1.1.1

Multiply $\frac{i}{2}$ by $\frac{i}{2}$ .

$\frac{i i}{2 \cdot 2} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.1.2

Raise $i$ to the power of $1$ .

$\frac{i^{1} i}{2 \cdot 2} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.1.3

Raise $i$ to the power of $1$ .

$\frac{i^{1} i^{1}}{2 \cdot 2} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{i^{1 + 1}}{2 \cdot 2} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.1.5

Add $1$ and $1$ .

$\frac{i^{2}}{2 \cdot 2} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.1.6

Multiply $2$ by $2$ .

$\frac{i^{2}}{4} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.2

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 1}{4} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.3

Move the negative in front of the fraction.

$- \frac{1}{4} + \frac{i}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.4

Multiply $\frac{i}{2} (- \frac{1}{2})$ .

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Step 1.5.1.4.1

Multiply $\frac{i}{2}$ by $\frac{1}{2}$ .

$- \frac{1}{4} - \frac{i}{2 \cdot 2} - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.4.2

Multiply $2$ by $2$ .

$- \frac{1}{4} - \frac{i}{4} - \frac{1}{2} \cdot \frac{i}{2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.5

Multiply $- \frac{1}{2} \cdot \frac{i}{2}$ .

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Step 1.5.1.5.1

Multiply $\frac{i}{2}$ by $\frac{1}{2}$ .

$- \frac{1}{4} - \frac{i}{4} - \frac{i}{2 \cdot 2} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.5.2

Multiply $2$ by $2$ .

$- \frac{1}{4} - \frac{i}{4} - \frac{i}{4} - \frac{1}{2} (- \frac{1}{2}) + \frac{i - 1}{2}$

Step 1.5.1.6

Multiply $- \frac{1}{2} (- \frac{1}{2})$ .

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Step 1.5.1.6.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{4} - \frac{i}{4} - \frac{i}{4} + 1 (\frac{1}{2}) \frac{1}{2} + \frac{i - 1}{2}$

Step 1.5.1.6.2

Multiply $\frac{1}{2}$ by $1$ .

$- \frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{1}{2} \cdot \frac{1}{2} + \frac{i - 1}{2}$

Step 1.5.1.6.3

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$- \frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{1}{2 \cdot 2} + \frac{i - 1}{2}$

Step 1.5.1.6.4

Multiply $2$ by $2$ .

$- \frac{1}{4} - \frac{i}{4} - \frac{i}{4} + \frac{1}{4} + \frac{i - 1}{2}$

Step 1.5.2

Add $- \frac{1}{4}$ and $\frac{1}{4}$ .

$0 - \frac{i}{4} - \frac{i}{4} + \frac{i - 1}{2}$

Step 1.5.3

Subtract $\frac{i}{4}$ from $0$ .

$- \frac{i}{4} - \frac{i}{4} + \frac{i - 1}{2}$

Step 1.5.4

Subtract $\frac{i}{4}$ from $- \frac{i}{4}$ .

$- 2 \frac{i}{4} + \frac{i - 1}{2}$

Step 1.6

Cancel the common factor of $2$ .

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Step 1.6.1

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{i}{4} + \frac{i - 1}{2}$

Step 1.6.2

Factor $2$ out of $4$ .

$2 \cdot - 1 \frac{i}{2 \cdot 2} + \frac{i - 1}{2}$

Step 1.6.3

Cancel the common factor.

$2 \cdot - 1 \frac{i}{2 \cdot 2} + \frac{i - 1}{2}$

Step 1.6.4

Rewrite the expression.

$- 1 \frac{i}{2} + \frac{i - 1}{2}$

Step 1.7

Rewrite $- 1 \frac{i}{2}$ as $- \frac{i}{2}$ .

$- \frac{i}{2} + \frac{i - 1}{2}$

Step 1.8

Split the fraction $\frac{i - 1}{2}$ into two fractions.

$- \frac{i}{2} + \frac{i}{2} + \frac{- 1}{2}$

Step 1.9

Move the negative in front of the fraction.

$- \frac{i}{2} + \frac{i}{2} - \frac{1}{2}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{- i + i - 1}{2}$

Step 2.2

Add $- i$ and $i$ .

$\frac{0 - 1}{2}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Subtract $1$ from $0$ .

$\frac{- 1}{2}$

Step 2.3.2

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$