Precalculus Examples

$(x - (5 + 4 i)) (x - (5 - 4 i))$

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.

$(x - 1 \cdot 5 - (4 i)) (x - (5 - 4 i))$

Step 1.1.2

Multiply $- 1$ by $5$ .

$(x - 5 - (4 i)) (x - (5 - 4 i))$

Step 1.1.3

Multiply $4$ by $- 1$ .

$(x - 5 - 4 i) (x - (5 - 4 i))$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$(x - 5 - 4 i) (x - 1 \cdot 5 - (- 4 i))$

Step 1.2.2

Multiply $- 1$ by $5$ .

$(x - 5 - 4 i) (x - 5 - (- 4 i))$

Step 1.2.3

Multiply $- 4$ by $- 1$ .

$(x - 5 - 4 i) (x - 5 + 4 i)$

Step 2

Expand $(x - 5 - 4 i) (x - 5 + 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 5 + x (4 i) - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \cdot - 5 + x (4 i) - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)$ .

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Step 3.1.1

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

$x \cdot x + x \cdot - 5 + 4 i x - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)$

Step 3.1.2

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

$x \cdot x + x \cdot - 5 + 0 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)$

Step 3.1.3

Add $x \cdot x + x \cdot - 5$ and $0$ .

$x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)$

Step 3.2.2

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)$

Step 3.2.3

Multiply $- 5$ by $- 5$ .

$x^{2} - 5 x - 5 x + 25 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)$

Step 3.2.4

Multiply $4$ by $- 5$ .

$x^{2} - 5 x - 5 x + 25 - 20 i - 4 i \cdot - 5 - 4 i (4 i)$

Step 3.2.5

Multiply $- 5$ by $- 4$ .

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 4 i (4 i)$

Step 3.2.6

Multiply $- 4 i (4 i)$ .

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Step 3.2.6.1

Multiply $4$ by $- 4$ .

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 i i$

Step 3.2.6.2

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

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 (i^{1} i)$

Step 3.2.6.3

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

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 (i^{1} i^{1})$

Step 3.2.6.4

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

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 i^{1 + 1}$

Step 3.2.6.5

Add $1$ and $1$ .

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 i^{2}$

Step 3.2.7

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

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 \cdot - 1$

Step 3.2.8

Multiply $- 16$ by $- 1$ .

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i + 16$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Combine the opposite terms in $x^{2} - 5 x - 5 x + 25 - 20 i + 20 i + 16$ .

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Step 3.3.1.1

Add $- 20 i$ and $20 i$ .

$x^{2} - 5 x - 5 x + 25 + 0 + 16$

Step 3.3.1.2

Add $x^{2} - 5 x - 5 x + 25$ and $0$ .

$x^{2} - 5 x - 5 x + 25 + 16$

Step 3.3.2

Subtract $5 x$ from $- 5 x$ .

$x^{2} - 10 x + 25 + 16$

Step 3.3.3

Add $25$ and $16$ .

$x^{2} - 10 x + 41$