Simplify (a+bi)^2

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

Step 1

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

$(a + b i) (a + b i)$

Step 2

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

Apply the distributive property.

$a (a + b i) + b i (a + b i)$

Apply the distributive property.

$a \cdot a + a (b i) + b i (a + b i)$

Apply the distributive property.

$a \cdot a + a (b i) + b i a + b i (b i)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $b$ .

$a^{2} + i a b + b i a + b \cdot b i i$

Multiply $b$ by $b$ .

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

Multiply $b^{2} i i$ .

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

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

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

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

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

$a^{2} + i a b + b i a + b^{2} i^{1 + 1}$

Add $1$ and $1$ .

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

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

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

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

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

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

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

Reorder $b$ and $a$ .

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

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

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