Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $x$ by $x$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Multiply $3 i (3 i)$ .

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Step 1.3.1.3.1

Multiply $3$ by $3$ .

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

Step 1.3.1.3.2

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

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

Step 1.3.1.3.3

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

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

Step 1.3.1.3.4

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

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

Step 1.3.1.3.5

Add $1$ and $1$ .

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

Step 1.3.1.4

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

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

Step 1.3.1.5

Multiply $9$ by $- 1$ .

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

Step 1.3.2

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

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Step 1.3.2.1

Move $i$ .

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

Step 1.3.2.2

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

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.2

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

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Step 1.6.1.2.1

Move $x$ .

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

Step 1.6.1.2.2

Multiply $x$ by $x$ .

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

Step 1.6.1.3

Multiply $2$ by $2$ .

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

Step 1.6.1.4

Multiply $- 3$ by $2$ .

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

Step 1.6.1.5

Multiply $2$ by $- 3$ .

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

Step 1.6.1.6

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

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Step 1.6.1.6.1

Multiply $- 3$ by $- 3$ .

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

Step 1.6.1.6.2

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

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

Step 1.6.1.6.3

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

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

Step 1.6.1.6.4

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

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

Step 1.6.1.6.5

Add $1$ and $1$ .

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

Step 1.6.1.7

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

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

Step 1.6.1.8

Multiply $9$ by $- 1$ .

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

Step 1.6.2

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

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Step 1.6.2.1

Move $i$ .

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

Step 1.6.2.2

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

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

Step 1.7

Apply the distributive property.

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

Step 1.8

Simplify.

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Step 1.8.1

Multiply $4$ by $- 1$ .

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

Step 1.8.2

Multiply $- 12$ by $- 1$ .

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

Step 1.8.3

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.

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Step 2.1

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

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Step 2.1.1

Add $- 9$ and $9$ .

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

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

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