Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

$(2 x - i) (2 x - i) - (2 x - i) (2 x + 3 i)$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$2 x (2 x - i) - i (2 x - i) - (2 x - i) (2 x + 3 i)$

Step 1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x (- i) - i (2 x - i) - (2 x - i) (2 x + 3 i)$

Step 1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x (- i) - i (2 x) - i (- i) - (2 x - i) (2 x + 3 i)$

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

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

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

Step 1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.1.4

Multiply $- 1$ by $2$ .

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

Step 1.3.1.5

Multiply $2$ by $- 1$ .

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

Step 1.3.1.6

Multiply $- i (- i)$ .

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Step 1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$4 x^{2} - 2 x i - 2 i x + 1 i i - (2 x - i) (2 x + 3 i)$

Step 1.3.1.6.2

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

$4 x^{2} - 2 x i - 2 i x + 1 (i^{1} i) - (2 x - i) (2 x + 3 i)$

Step 1.3.1.6.3

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

$4 x^{2} - 2 x i - 2 i x + 1 (i^{1} i^{1}) - (2 x - i) (2 x + 3 i)$

Step 1.3.1.6.4

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

$4 x^{2} - 2 x i - 2 i x + 1 i^{1 + 1} - (2 x - i) (2 x + 3 i)$

Step 1.3.1.6.5

Add $1$ and $1$ .

$4 x^{2} - 2 x i - 2 i x + 1 i^{2} - (2 x - i) (2 x + 3 i)$

Step 1.3.1.6.6

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

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

Step 1.3.1.7

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

$4 x^{2} - 2 x i - 2 i x - 1 - (2 x - i) (2 x + 3 i)$

Step 1.3.2

Subtract $2 i x$ from $- 2 x i$ .

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Step 1.3.2.1

Move $i$ .

$4 x^{2} - 2 x i - 2 x i - 1 - (2 x - i) (2 x + 3 i)$

Step 1.3.2.2

Subtract $2 x i$ from $- 2 x i$ .

$4 x^{2} - 4 x i - 1 - (2 x - i) (2 x + 3 i)$

Step 1.4

Apply the distributive property.

$4 x^{2} - 4 x i - 1 + (- (2 x) - - i) (2 x + 3 i)$

Step 1.5

Multiply $2$ by $- 1$ .

$4 x^{2} - 4 x i - 1 + (- 2 x - - i) (2 x + 3 i)$

Step 1.6

Multiply $- 1$ by $- 1$ .

$4 x^{2} - 4 x i - 1 + (- 2 x + 1 i) (2 x + 3 i)$

Step 1.7

Multiply $i$ by $1$ .

$4 x^{2} - 4 x i - 1 + (- 2 x + i) (2 x + 3 i)$

Step 1.8

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

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Step 1.8.1

Apply the distributive property.

$4 x^{2} - 4 x i - 1 - 2 x (2 x + 3 i) + i (2 x + 3 i)$

Step 1.8.2

Apply the distributive property.

$4 x^{2} - 4 x i - 1 - 2 x (2 x) - 2 x (3 i) + i (2 x + 3 i)$

Step 1.8.3

Apply the distributive property.

$4 x^{2} - 4 x i - 1 - 2 x (2 x) - 2 x (3 i) + i (2 x) + i (3 i)$

Step 1.9

Simplify and combine like terms.

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Rewrite using the commutative property of multiplication.

$4 x^{2} - 4 x i - 1 - 2 \cdot 2 x \cdot x - 2 x (3 i) + i (2 x) + i (3 i)$

Step 1.9.1.2

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

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Step 1.9.1.2.1

Move $x$ .

$4 x^{2} - 4 x i - 1 - 2 \cdot 2 (x \cdot x) - 2 x (3 i) + i (2 x) + i (3 i)$

Step 1.9.1.2.2

Multiply $x$ by $x$ .

$4 x^{2} - 4 x i - 1 - 2 \cdot 2 x^{2} - 2 x (3 i) + i (2 x) + i (3 i)$

Step 1.9.1.3

Multiply $- 2$ by $2$ .

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 2 x (3 i) + i (2 x) + i (3 i)$

Step 1.9.1.4

Multiply $3$ by $- 2$ .

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + i (2 x) + i (3 i)$

Step 1.9.1.5

Move $2$ to the left of $i$ .

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + 2 \cdot i x + i (3 i)$

Step 1.9.1.6

Multiply $i (3 i)$ .

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Step 1.9.1.6.1

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

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + 2 i x + 3 (i^{1} i)$

Step 1.9.1.6.2

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

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + 2 i x + 3 (i^{1} i^{1})$

Step 1.9.1.6.3

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

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + 2 i x + 3 i^{1 + 1}$

Step 1.9.1.6.4

Add $1$ and $1$ .

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + 2 i x + 3 i^{2}$

Step 1.9.1.7

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

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + 2 i x + 3 \cdot - 1$

Step 1.9.1.8

Multiply $3$ by $- 1$ .

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + 2 i x - 3$

Step 1.9.2

Add $- 6 x i$ and $2 i x$ .

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Step 1.9.2.1

Move $i$ .

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 6 x i + 2 x i - 3$

Step 1.9.2.2

Add $- 6 x i$ and $2 x i$ .

$4 x^{2} - 4 x i - 1 - 4 x^{2} - 4 x i - 3$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $4 x^{2} - 4 x i - 1 - 4 x^{2} - 4 x i - 3$ .

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Step 2.1.1

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

$- 4 x i - 1 + 0 - 4 x i - 3$

Step 2.1.2

Add $- 4 x i - 1$ and $0$ .

$- 4 x i - 1 - 4 x i - 3$

Step 2.2

Subtract $4 x i$ from $- 4 x i$ .

$- 8 x i - 1 - 3$

Step 2.3

Subtract $3$ from $- 1$ .

$- 8 x i - 4$