Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Expand $(1 - 2 i) (1 - 2 i)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

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

Step 1.3.1.2

Multiply $- 2 i$ by $1$ .

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

Step 1.3.1.3

Multiply $- 2$ by $1$ .

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

Step 1.3.1.4

Multiply $- 2 i (- 2 i)$ .

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Step 1.3.1.4.1

Multiply $- 2$ by $- 2$ .

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

Step 1.3.1.4.2

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

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

Step 1.3.1.4.3

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

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

Step 1.3.1.4.4

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

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

Step 1.3.1.4.5

Add $1$ and $1$ .

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

Step 1.3.1.5

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

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

Step 1.3.1.6

Multiply $4$ by $- 1$ .

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

Step 1.3.2

Subtract $4$ from $1$ .

$- 3 - 2 i - 2 i - {(1 + 2 i)}^{2}$

Step 1.3.3

Subtract $2 i$ from $- 2 i$ .

$- 3 - 4 i - {(1 + 2 i)}^{2}$

Step 1.4

Rewrite ${(1 + 2 i)}^{2}$ as $(1 + 2 i) (1 + 2 i)$ .

$- 3 - 4 i - ((1 + 2 i) (1 + 2 i))$

Step 1.5

Expand $(1 + 2 i) (1 + 2 i)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$- 3 - 4 i - (1 (1 + 2 i) + 2 i (1 + 2 i))$

Step 1.5.2

Apply the distributive property.

$- 3 - 4 i - (1 \cdot 1 + 1 (2 i) + 2 i (1 + 2 i))$

Step 1.5.3

Apply the distributive property.

$- 3 - 4 i - (1 \cdot 1 + 1 (2 i) + 2 i \cdot 1 + 2 i (2 i))$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $1$ by $1$ .

$- 3 - 4 i - (1 + 1 (2 i) + 2 i \cdot 1 + 2 i (2 i))$

Step 1.6.1.2

Multiply $2 i$ by $1$ .

$- 3 - 4 i - (1 + 2 i + 2 i \cdot 1 + 2 i (2 i))$

Step 1.6.1.3

Multiply $2$ by $1$ .

$- 3 - 4 i - (1 + 2 i + 2 i + 2 i (2 i))$

Step 1.6.1.4

Multiply $2 i (2 i)$ .

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Step 1.6.1.4.1

Multiply $2$ by $2$ .

$- 3 - 4 i - (1 + 2 i + 2 i + 4 i i)$

Step 1.6.1.4.2

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

$- 3 - 4 i - (1 + 2 i + 2 i + 4 (i^{1} i))$

Step 1.6.1.4.3

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

$- 3 - 4 i - (1 + 2 i + 2 i + 4 (i^{1} i^{1}))$

Step 1.6.1.4.4

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

$- 3 - 4 i - (1 + 2 i + 2 i + 4 i^{1 + 1})$

Step 1.6.1.4.5

Add $1$ and $1$ .

$- 3 - 4 i - (1 + 2 i + 2 i + 4 i^{2})$

Step 1.6.1.5

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

$- 3 - 4 i - (1 + 2 i + 2 i + 4 \cdot - 1)$

Step 1.6.1.6

Multiply $4$ by $- 1$ .

$- 3 - 4 i - (1 + 2 i + 2 i - 4)$

Step 1.6.2

Subtract $4$ from $1$ .

$- 3 - 4 i - (- 3 + 2 i + 2 i)$

Step 1.6.3

Add $2 i$ and $2 i$ .

$- 3 - 4 i - (- 3 + 4 i)$

Step 1.7

Apply the distributive property.

$- 3 - 4 i - - 3 - (4 i)$

Step 1.8

Multiply $- 1$ by $- 3$ .

$- 3 - 4 i + 3 - (4 i)$

Step 1.9

Multiply $4$ by $- 1$ .

$- 3 - 4 i + 3 - 4 i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- 3$ and $3$ .

$0 - 4 i - 4 i$

Step 2.2

Subtract $4 i$ from $0$ .

$- 4 i - 4 i$

Step 2.3

Subtract $4 i$ from $- 4 i$ .

$- 8 i$