Simplify (x+3i)^2-(2x-3i)^2

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

Step 1

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

$x \cdot x + x (3 i) + 3 i (x + 3 i) - {(2 x - 3 i)}^{2}$

Apply the distributive property.

$x \cdot x + x (3 i) + 3 i x + 3 i (3 i) - {(2 x - 3 i)}^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $3$ to the left of $x$ .

$x^{2} + 3 \cdot x i + 3 i x + 3 i (3 i) - {(2 x - 3 i)}^{2}$

Multiply $3 i (3 i)$ .

Multiply $3$ by $3$ .

$x^{2} + 3 x i + 3 i x + 9 i i - {(2 x - 3 i)}^{2}$

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

$x^{2} + 3 x i + 3 i x + 9 (i^{1} i) - {(2 x - 3 i)}^{2}$

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

$x^{2} + 3 x i + 3 i x + 9 (i^{1} i^{1}) - {(2 x - 3 i)}^{2}$

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

$x^{2} + 3 x i + 3 i x + 9 i^{1 + 1} - {(2 x - 3 i)}^{2}$

Add $1$ and $1$ .

$x^{2} + 3 x i + 3 i x + 9 i^{2} - {(2 x - 3 i)}^{2}$

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

$x^{2} + 3 x i + 3 i x + 9 \cdot - 1 - {(2 x - 3 i)}^{2}$

Multiply $9$ by $- 1$ .

$x^{2} + 3 x i + 3 i x - 9 - {(2 x - 3 i)}^{2}$

Add $3 x i$ and $3 i x$ .

Move $i$ .

$x^{2} + 3 x i + 3 x i - 9 - {(2 x - 3 i)}^{2}$

Add $3 x i$ and $3 x i$ .

$x^{2} + 6 x i - 9 - {(2 x - 3 i)}^{2}$

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

$x^{2} + 6 x i - 9 - ((2 x - 3 i) (2 x - 3 i))$

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

Apply the distributive property.

$x^{2} + 6 x i - 9 - (2 x (2 x - 3 i) - 3 i (2 x - 3 i))$

Apply the distributive property.

$x^{2} + 6 x i - 9 - (2 x (2 x) + 2 x (- 3 i) - 3 i (2 x - 3 i))$

Apply the distributive property.

$x^{2} + 6 x i - 9 - (2 x (2 x) + 2 x (- 3 i) - 3 i (2 x) - 3 i (- 3 i))$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{2} + 6 x i - 9 - (2 \cdot 2 x \cdot x + 2 x (- 3 i) - 3 i (2 x) - 3 i (- 3 i))$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{2} + 6 x i - 9 - (2 \cdot 2 (x \cdot x) + 2 x (- 3 i) - 3 i (2 x) - 3 i (- 3 i))$

Multiply $x$ by $x$ .

$x^{2} + 6 x i - 9 - (2 \cdot 2 x^{2} + 2 x (- 3 i) - 3 i (2 x) - 3 i (- 3 i))$

Multiply $2$ by $2$ .

$x^{2} + 6 x i - 9 - (4 x^{2} + 2 x (- 3 i) - 3 i (2 x) - 3 i (- 3 i))$

Multiply $- 3$ by $2$ .

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 3 i (2 x) - 3 i (- 3 i))$

Multiply $2$ by $- 3$ .

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 i x - 3 i (- 3 i))$

Multiply $- 3 i (- 3 i)$ .

Multiply $- 3$ by $- 3$ .

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 i x + 9 i i)$

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

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 i x + 9 (i^{1} i))$

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

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 i x + 9 (i^{1} i^{1}))$

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

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 i x + 9 i^{1 + 1})$

Add $1$ and $1$ .

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 i x + 9 i^{2})$

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

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 i x + 9 \cdot - 1)$

Multiply $9$ by $- 1$ .

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 i x - 9)$

Subtract $6 i x$ from $- 6 x i$ .

Move $i$ .

$x^{2} + 6 x i - 9 - (4 x^{2} - 6 x i - 6 x i - 9)$

Subtract $6 x i$ from $- 6 x i$ .

$x^{2} + 6 x i - 9 - (4 x^{2} - 12 x i - 9)$

Apply the distributive property.

$x^{2} + 6 x i - 9 - (4 x^{2}) - (- 12 x i) - - 9$

Simplify.

Multiply $4$ by $- 1$ .

$x^{2} + 6 x i - 9 - 4 x^{2} - (- 12 x i) - - 9$

Multiply $- 12$ by $- 1$ .

$x^{2} + 6 x i - 9 - 4 x^{2} + 12 (x i) - - 9$

Multiply $- 1$ by $- 9$ .

$x^{2} + 6 x i - 9 - 4 x^{2} + 12 (x i) + 9$

$x^{2} + 6 x i - 9 - 4 x^{2} + 12 x i + 9$

Step 2

Simplify by adding terms.

Combine the opposite terms in $x^{2} + 6 x i - 9 - 4 x^{2} + 12 x i + 9$ .

Add $- 9$ and $9$ .

$x^{2} + 6 x i - 4 x^{2} + 12 x i + 0$

Add $x^{2} + 6 x i - 4 x^{2} + 12 x i$ and $0$ .

$x^{2} + 6 x i - 4 x^{2} + 12 x i$

Subtract $4 x^{2}$ from $x^{2}$ .

$- 3 x^{2} + 6 x i + 12 x i$

Add $6 x i$ and $12 x i$ .

$- 3 x^{2} + 18 x i$

Simplify (x+3i)^2-(2x-3i)^2

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