Finite Math Examples

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

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.

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

Step 1.1.2

Multiply $- 1$ by $3$ .

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Multiply $- 1$ by $3$ .

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

Step 1.2.3

Multiply $- 1$ by $- 1$ .

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

Step 1.2.4

Multiply $i$ by $1$ .

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

Step 1.2.5

Expand $(- 3 + i) (x - 4)$ using the FOIL Method.

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Step 1.2.5.1

Apply the distributive property.

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

Step 1.2.5.2

Apply the distributive property.

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

Step 1.2.5.3

Apply the distributive property.

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

Step 1.2.6

Simplify each term.

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Step 1.2.6.1

Multiply $- 3$ by $- 4$ .

$(x - 3 - i) (x - 3 x + 12 + i x + i \cdot - 4)$

Step 1.2.6.2

Move $- 4$ to the left of $i$ .

$(x - 3 - i) (x - 3 x + 12 + i x - 4 i)$

Step 1.3

Subtract $3 x$ from $x$ .

$(x - 3 - i) (- 2 x + 12 + i x - 4 i)$

Step 2

Expand $(x - 3 - i) (- 2 x + 12 + i x - 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$x (- 2 x) + x \cdot 12 + x (i x) + x (- 4 i) - 3 (- 2 x) - 3 \cdot 12 - 3 (i x) - 3 (- 4 i) - i (- 2 x) - i \cdot 12 - i (i x) - i (- 4 i)$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$- 2 x \cdot x + x \cdot 12 + x (i x) + x (- 4 i) - 3 (- 2 x) - 3 \cdot 12 - 3 (i x) - 3 (- 4 i) - i (- 2 x) - i \cdot 12 - i (i x) - i (- 4 i)$

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

$- 2 (x \cdot x) + x \cdot 12 + x (i x) + x (- 4 i) - 3 (- 2 x) - 3 \cdot 12 - 3 (i x) - 3 (- 4 i) - i (- 2 x) - i \cdot 12 - i (i x) - i (- 4 i)$

Step 3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3

Move $12$ to the left of $x$ .

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

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

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Step 3.1.5.1

Move $x$ .

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

Step 3.1.5.2

Multiply $x$ by $x$ .

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

Step 3.1.6

Multiply $- 2$ by $- 3$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 3 \cdot 12 - 3 (i x) - 3 (- 4 i) - i (- 2 x) - i \cdot 12 - i (i x) - i (- 4 i)$

Step 3.1.7

Multiply $- 3$ by $12$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 (i x) - 3 (- 4 i) - i (- 2 x) - i \cdot 12 - i (i x) - i (- 4 i)$

Step 3.1.8

Multiply $- 4$ by $- 3$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i - i (- 2 x) - i \cdot 12 - i (i x) - i (- 4 i)$

Step 3.1.9

Multiply $- 2$ by $- 1$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - i \cdot 12 - i (i x) - i (- 4 i)$

Step 3.1.10

Multiply $12$ by $- 1$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i - i (i x) - i (- 4 i)$

Step 3.1.11

Multiply $- i (i x)$ .

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Step 3.1.11.1

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

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i - (i^{1} i) x - i (- 4 i)$

Step 3.1.11.2

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

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i - (i^{1} i^{1}) x - i (- 4 i)$

Step 3.1.11.3

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

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i - i^{1 + 1} x - i (- 4 i)$

Step 3.1.11.4

Add $1$ and $1$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i - i^{2} x - i (- 4 i)$

Step 3.1.12

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

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i - - 1 x - i (- 4 i)$

Step 3.1.13

Multiply $- 1$ by $- 1$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + 1 x - i (- 4 i)$

Step 3.1.14

Multiply $x$ by $1$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x - i (- 4 i)$

Step 3.1.15

Multiply $- i (- 4 i)$ .

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Step 3.1.15.1

Multiply $- 4$ by $- 1$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x + 4 i i$

Step 3.1.15.2

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

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x + 4 (i^{1} i)$

Step 3.1.15.3

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

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x + 4 (i^{1} i^{1})$

Step 3.1.15.4

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

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x + 4 i^{1 + 1}$

Step 3.1.15.5

Add $1$ and $1$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x + 4 i^{2}$

Step 3.1.16

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

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x + 4 \cdot - 1$

Step 3.1.17

Multiply $4$ by $- 1$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x - 4$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Combine the opposite terms in $- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 12 i + 2 i x - 12 i + x - 4$ .

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Step 3.2.1.1

Subtract $12 i$ from $12 i$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 2 i x + 0 + x - 4$

Step 3.2.1.2

Add $- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 2 i x$ and $0$ .

$- 2 x^{2} + 12 x + i x^{2} + x (- 4 i) + 6 x - 36 - 3 i x + 2 i x + x - 4$

Step 3.2.2

Add $12 x$ and $6 x$ .

$- 2 x^{2} + i x^{2} + x (- 4 i) + 18 x - 36 - 3 i x + 2 i x + x - 4$

Step 3.2.3

Move $i$ .

$- 2 x^{2} + i x^{2} - 4 \cdot x i - 3 x i + 18 x - 36 + 2 i x + x - 4$

Step 3.2.4

Subtract $3 x i$ from $- 4 x i$ .

$- 2 x^{2} + i x^{2} - 7 x i + 18 x - 36 + 2 i x + x - 4$

Step 4

Add $- 7 x i$ and $2 i x$ .

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Step 4.1

Move $i$ .

$- 2 x^{2} + i x^{2} - 7 x i + 2 x i + 18 x - 36 + x - 4$

Step 4.2

Add $- 7 x i$ and $2 x i$ .

$- 2 x^{2} + i x^{2} - 5 x i + 18 x - 36 + x - 4$

Step 5

Add $18 x$ and $x$ .

$- 2 x^{2} + i x^{2} - 5 x i + 19 x - 36 - 4$

Step 6

Subtract $4$ from $- 36$ .

$- 2 x^{2} + i x^{2} - 5 x i + 19 x - 40$