Algebra Examples

${(a + b i)}^{2} - {(a - b i)}^{2}$

Step 1

Simplify each term.

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Step 1.1

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

$(a + b i) (a + b i) - {(a - b i)}^{2}$

Step 1.2

Expand $(a + b i) (a + b i)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$a (a + b i) + b i (a + b i) - {(a - b i)}^{2}$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $a$ by $a$ .

$a^{2} + a (b i) + b i a + b i (b i) - {(a - b i)}^{2}$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$a^{2} + i a b + b i a + b i (b i) - {(a - b i)}^{2}$

Step 1.3.1.3

Multiply $b$ by $b$ by adding the exponents.

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Step 1.3.1.3.1

Move $b$ .

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

Step 1.3.1.3.2

Multiply $b$ by $b$ .

$a^{2} + i a b + b i a + b^{2} i i - {(a - b i)}^{2}$

Step 1.3.1.4

Multiply $b^{2} i i$ .

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Step 1.3.1.4.1

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

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

Step 1.3.1.4.2

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

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

Step 1.3.1.4.3

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

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

Step 1.3.1.4.4

Add $1$ and $1$ .

$a^{2} + i a b + b i a + b^{2} i^{2} - {(a - b i)}^{2}$

Step 1.3.1.5

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

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

Step 1.3.1.6

Move $- 1$ to the left of $b^{2}$ .

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

Step 1.3.1.7

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

$a^{2} + i a b + b i a - b^{2} - {(a - b i)}^{2}$

Step 1.3.2

Reorder $b$ and $a$ .

$a^{2} + a b i + a b i - b^{2} - {(a - b i)}^{2}$

Step 1.3.3

Add $a b i$ and $a b i$ .

$a^{2} + 2 a b i - b^{2} - {(a - b i)}^{2}$

Step 1.4

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

$a^{2} + 2 a b i - b^{2} - ((a - b i) (a - b i))$

Step 1.5

Expand $(a - b i) (a - b i)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$a^{2} + 2 a b i - b^{2} - (a (a - b i) - b i (a - b i))$

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

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 $a$ by $a$ .

$a^{2} + 2 a b i - b^{2} - (a^{2} + a (- b i) - b i a - b i (- b i))$

Step 1.6.1.2

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b i - b^{2} - (a^{2} - i a b - b i a - b i (- b i))$

Step 1.6.1.3

Multiply $b$ by $b$ by adding the exponents.

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Step 1.6.1.3.1

Move $b$ .

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

Step 1.6.1.3.2

Multiply $b$ by $b$ .

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

Step 1.6.1.4

Rewrite $- 1 i$ as $- i$ .

$a^{2} + 2 a b i - b^{2} - (a^{2} - i a b - b i a - b^{2} i (- i))$

Step 1.6.1.5

Multiply $- b^{2} i (- i)$ .

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Step 1.6.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.5.2

Multiply $b^{2}$ by $1$ .

$a^{2} + 2 a b i - b^{2} - (a^{2} - i a b - b i a + b^{2} i i)$

Step 1.6.1.5.3

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

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

Step 1.6.1.5.4

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

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

Step 1.6.1.5.5

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

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

Step 1.6.1.5.6

Add $1$ and $1$ .

$a^{2} + 2 a b i - b^{2} - (a^{2} - i a b - b i a + b^{2} i^{2})$

Step 1.6.1.6

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

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

Step 1.6.1.7

Move $- 1$ to the left of $b^{2}$ .

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

Step 1.6.1.8

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

$a^{2} + 2 a b i - b^{2} - (a^{2} - i a b - b i a - b^{2})$

Step 1.6.2

Move $b$ .

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

Step 1.6.3

Subtract $a b i$ from $- a b i$ .

$a^{2} + 2 a b i - b^{2} - (a^{2} - 2 a b i - b^{2})$

Step 1.7

Apply the distributive property.

$a^{2} + 2 a b i - b^{2} - a^{2} - (- 2 a b i) - - b^{2}$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $- 2$ by $- 1$ .

$a^{2} + 2 a b i - b^{2} - a^{2} + 2 (a b i) - - b^{2}$

Step 1.8.2

Multiply $- - b^{2}$ .

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Step 1.8.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.8.2.2

Multiply $b^{2}$ by $1$ .

$a^{2} + 2 a b i - b^{2} - a^{2} + 2 (a b i) + b^{2}$

Step 1.9

Remove parentheses.

$a^{2} + 2 a b i - b^{2} - a^{2} + 2 a b i + b^{2}$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $a^{2} + 2 a b i - b^{2} - a^{2} + 2 a b i + b^{2}$ .

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Step 2.1.1

Subtract $a^{2}$ from $a^{2}$ .

$2 a b i - b^{2} + 0 + 2 a b i + b^{2}$

Step 2.1.2

Add $2 a b i - b^{2}$ and $0$ .

$2 a b i - b^{2} + 2 a b i + b^{2}$

Step 2.1.3

Add $- b^{2}$ and $b^{2}$ .

$2 a b i + 2 a b i + 0$

Step 2.1.4

Add $2 a b i + 2 a b i$ and $0$ .

$2 a b i + 2 a b i$

Step 2.2

Add $2 a b i$ and $2 a b i$ .

$4 a b i$