Basic Math Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $z \cdot z + z \cdot - 3 + z (2 i) - 3 z - 3 \cdot - 3 - 3 (2 i) - 2 i z - 2 i \cdot - 3 - 2 i (2 i)$ .

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Step 2.1.1

Reorder the factors in the terms $z (2 i)$ and $- 2 i z$ .

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

Step 2.1.2

Subtract $2 i z$ from $2 i z$ .

$z \cdot z + z \cdot - 3 + 0 - 3 z - 3 \cdot - 3 - 3 (2 i) - 2 i \cdot - 3 - 2 i (2 i)$

Step 2.1.3

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $z$ by $z$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $- 3$ by $- 3$ .

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

Step 2.2.4

Multiply $2$ by $- 3$ .

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

Step 2.2.5

Multiply $- 3$ by $- 2$ .

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

Step 2.2.6

Multiply $- 2 i (2 i)$ .

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Step 2.2.6.1

Multiply $2$ by $- 2$ .

$z^{2} - 3 z - 3 z + 9 - 6 i + 6 i - 4 i i$

Step 2.2.6.2

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

$z^{2} - 3 z - 3 z + 9 - 6 i + 6 i - 4 (i^{1} i)$

Step 2.2.6.3

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

$z^{2} - 3 z - 3 z + 9 - 6 i + 6 i - 4 (i^{1} i^{1})$

Step 2.2.6.4

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

$z^{2} - 3 z - 3 z + 9 - 6 i + 6 i - 4 i^{1 + 1}$

Step 2.2.6.5

Add $1$ and $1$ .

$z^{2} - 3 z - 3 z + 9 - 6 i + 6 i - 4 i^{2}$

Step 2.2.7

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

$z^{2} - 3 z - 3 z + 9 - 6 i + 6 i - 4 \cdot - 1$

Step 2.2.8

Multiply $- 4$ by $- 1$ .

$z^{2} - 3 z - 3 z + 9 - 6 i + 6 i + 4$

Step 2.3

Simplify by adding terms.

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Step 2.3.1

Combine the opposite terms in $z^{2} - 3 z - 3 z + 9 - 6 i + 6 i + 4$ .

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Step 2.3.1.1

Add $- 6 i$ and $6 i$ .

$z^{2} - 3 z - 3 z + 9 + 0 + 4$

Step 2.3.1.2

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

$z^{2} - 3 z - 3 z + 9 + 4$

Step 2.3.2

Subtract $3 z$ from $- 3 z$ .

$z^{2} - 6 z + 9 + 4$

Step 2.3.3

Add $9$ and $4$ .

$z^{2} - 6 z + 13$