Simplify (2+3i)^2

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

Step 1

Rewrite ${(2 + 3 i)}^{2}$ as $(2 + 3 i) (2 + 3 i)$ .

$(2 + 3 i) (2 + 3 i)$

Step 2

Expand $(2 + 3 i) (2 + 3 i)$ using the FOIL Method.

Apply the distributive property.

$2 (2 + 3 i) + 3 i (2 + 3 i)$

Apply the distributive property.

$2 \cdot 2 + 2 (3 i) + 3 i (2 + 3 i)$

Apply the distributive property.

$2 \cdot 2 + 2 (3 i) + 3 i \cdot 2 + 3 i (3 i)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $3$ by $2$ .

$4 + 6 i + 3 i \cdot 2 + 3 i (3 i)$

Multiply $2$ by $3$ .

$4 + 6 i + 6 i + 3 i (3 i)$

Multiply $3 i (3 i)$ .

Multiply $3$ by $3$ .

$4 + 6 i + 6 i + 9 i i$

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

$4 + 6 i + 6 i + 9 (i^{1} i)$

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

$4 + 6 i + 6 i + 9 (i^{1} i^{1})$

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

$4 + 6 i + 6 i + 9 i^{1 + 1}$

Add $1$ and $1$ .

$4 + 6 i + 6 i + 9 i^{2}$

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

$4 + 6 i + 6 i + 9 \cdot - 1$

Multiply $9$ by $- 1$ .

$4 + 6 i + 6 i - 9$

Subtract $9$ from $4$ .

$- 5 + 6 i + 6 i$

Add $6 i$ and $6 i$ .

$- 5 + 12 i$