Precalculus Examples

$(x - (3 - 4 i)) (x - (3 + 4 i))$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$(x - 1 \cdot 3 - (- 4 i)) (x - (3 + 4 i))$

Step 1.1.2

Multiply $- 1$ by $3$ .

$(x - 3 - (- 4 i)) (x - (3 + 4 i))$

Step 1.1.3

Multiply $- 4$ by $- 1$ .

$(x - 3 + 4 i) (x - (3 + 4 i))$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$(x - 3 + 4 i) (x - 1 \cdot 3 - (4 i))$

Step 1.2.2

Multiply $- 1$ by $3$ .

$(x - 3 + 4 i) (x - 3 - (4 i))$

Step 1.2.3

Multiply $4$ by $- 1$ .

$(x - 3 + 4 i) (x - 3 - 4 i)$

Step 2

Expand $(x - 3 + 4 i) (x - 3 - 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 3 + x (- 4 i) - 3 x - 3 \cdot - 3 - 3 (- 4 i) + 4 i x + 4 i \cdot - 3 + 4 i (- 4 i)$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \cdot - 3 + x (- 4 i) - 3 x - 3 \cdot - 3 - 3 (- 4 i) + 4 i x + 4 i \cdot - 3 + 4 i (- 4 i)$ .

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Step 3.1.1

Reorder the factors in the terms $x (- 4 i)$ and $4 i x$ .

$x \cdot x + x \cdot - 3 - 4 i x - 3 x - 3 \cdot - 3 - 3 (- 4 i) + 4 i x + 4 i \cdot - 3 + 4 i (- 4 i)$

Step 3.1.2

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

$x \cdot x + x \cdot - 3 + 0 - 3 x - 3 \cdot - 3 - 3 (- 4 i) + 4 i \cdot - 3 + 4 i (- 4 i)$

Step 3.1.3

Add $x \cdot x + x \cdot - 3$ and $0$ .

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 (- 4 i) + 4 i \cdot - 3 + 4 i (- 4 i)$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $- 3$ by $- 3$ .

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

Step 3.2.4

Multiply $- 4$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 + 12 i + 4 i \cdot - 3 + 4 i (- 4 i)$

Step 3.2.5

Multiply $- 3$ by $4$ .

$x^{2} - 3 x - 3 x + 9 + 12 i - 12 i + 4 i (- 4 i)$

Step 3.2.6

Multiply $4 i (- 4 i)$ .

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Step 3.2.6.1

Multiply $- 4$ by $4$ .

$x^{2} - 3 x - 3 x + 9 + 12 i - 12 i - 16 i i$

Step 3.2.6.2

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

$x^{2} - 3 x - 3 x + 9 + 12 i - 12 i - 16 (i^{1} i)$

Step 3.2.6.3

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

$x^{2} - 3 x - 3 x + 9 + 12 i - 12 i - 16 (i^{1} i^{1})$

Step 3.2.6.4

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

$x^{2} - 3 x - 3 x + 9 + 12 i - 12 i - 16 i^{1 + 1}$

Step 3.2.6.5

Add $1$ and $1$ .

$x^{2} - 3 x - 3 x + 9 + 12 i - 12 i - 16 i^{2}$

Step 3.2.7

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

$x^{2} - 3 x - 3 x + 9 + 12 i - 12 i - 16 \cdot - 1$

Step 3.2.8

Multiply $- 16$ by $- 1$ .

$x^{2} - 3 x - 3 x + 9 + 12 i - 12 i + 16$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Combine the opposite terms in $x^{2} - 3 x - 3 x + 9 + 12 i - 12 i + 16$ .

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Step 3.3.1.1

Subtract $12 i$ from $12 i$ .

$x^{2} - 3 x - 3 x + 9 + 0 + 16$

Step 3.3.1.2

Add $x^{2} - 3 x - 3 x + 9$ and $0$ .

$x^{2} - 3 x - 3 x + 9 + 16$

Step 3.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 6 x + 9 + 16$

Step 3.3.3

Add $9$ and $16$ .

$x^{2} - 6 x + 25$