Finite Math Examples

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

Step 1

Simplify $z (2 - 3 i)$ .

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Step 1.1

Rewrite.

$0 + 0 + z (2 - 3 i) = z (3 - 2 i) + 3 - 2 i$

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Simplify the expression.

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Step 1.4.1

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

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

Step 1.4.2

Reorder factors in $2 z + z (- 3 i)$ .

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

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

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

Step 2.2

Reorder factors in $3 z + z (- 2 i) + 3 - 2 i$ .

$2 z - 3 z i = 3 z - 2 z i + 3 - 2 i$

Step 3

Move all terms containing $z$ to the left side of the equation.

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Step 3.1

Subtract $3 z$ from both sides of the equation.

$2 z - 3 z i - 3 z = - 2 z i + 3 - 2 i$

Step 3.2

Add $2 z i$ to both sides of the equation.

$2 z - 3 z i - 3 z + 2 z i = 3 - 2 i$

Step 3.3

Subtract $3 z$ from $2 z$ .

$- 3 z i - z + 2 z i = 3 - 2 i$

Step 3.4

Add $- 3 z i$ and $2 z i$ .

$- z i - z = 3 - 2 i$

Step 4

Factor $- z$ out of $- z i - z$ .

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Step 4.1

Factor $- z$ out of $- z$ .

$- z i - z \cdot 1 = 3 - 2 i$

Step 4.2

Factor $- z$ out of $- z (i) - z (1)$ .

$- z (i + 1) = 3 - 2 i$

Step 5

Divide each term in $- z (i + 1) = 3 - 2 i$ by $- i - 1$ and simplify.

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Step 5.1

Divide each term in $- z (i + 1) = 3 - 2 i$ by $- i - 1$ .

$\frac{- z (i + 1)}{- i - 1} = \frac{3}{- i - 1} + \frac{- 2 i}{- i - 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $i + 1$ and $- i - 1$ .

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Step 5.2.1.1

Factor $- 1$ out of $i$ .

$\frac{- z (- 1 (- i) + 1)}{- i - 1} = \frac{3}{- i - 1} + \frac{- 2 i}{- i - 1}$

Step 5.2.1.2

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- z (- 1 (- i) - 1 (- 1))}{- i - 1} = \frac{3}{- i - 1} + \frac{- 2 i}{- i - 1}$

Step 5.2.1.3

Factor $- 1$ out of $- 1 (- i) - 1 (- 1)$ .

$\frac{- z (- 1 (- i - 1))}{- i - 1} = \frac{3}{- i - 1} + \frac{- 2 i}{- i - 1}$

Step 5.2.1.4

Cancel the common factor.

$\frac{- z (- 1 (- i - 1))}{- i - 1} = \frac{3}{- i - 1} + \frac{- 2 i}{- i - 1}$

Step 5.2.1.5

Divide $- z \cdot (- 1)$ by $1$ .

$- z \cdot (- 1) = \frac{3}{- i - 1} + \frac{- 2 i}{- i - 1}$

Step 5.2.2

Multiply $- z (- 1)$ .

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Step 5.2.2.1

Multiply $- 1$ by $- 1$ .

$1 z = \frac{3}{- i - 1} + \frac{- 2 i}{- i - 1}$

Step 5.2.2.2

Multiply $z$ by $1$ .

$z = \frac{3}{- i - 1} + \frac{- 2 i}{- i - 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Combine the numerators over the common denominator.

$z = \frac{3 - 2 i}{- i - 1}$

Step 5.3.2

Multiply the numerator and denominator of $\frac{3 - 2 i}{- 1 - i}$ by the conjugate of $- 1 - i$ to make the denominator real.

$z = \frac{3 - 2 i}{- 1 - i} \cdot \frac{- 1 + i}{- 1 + i}$

Step 5.3.3

Multiply.

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Step 5.3.3.1

Combine.

$z = \frac{(3 - 2 i) (- 1 + i)}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Expand $(3 - 2 i) (- 1 + i)$ using the FOIL Method.

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Step 5.3.3.2.1.1

Apply the distributive property.

$z = \frac{3 (- 1 + i) - 2 i (- 1 + i)}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.1.2

Apply the distributive property.

$z = \frac{3 \cdot - 1 + 3 i - 2 i (- 1 + i)}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.1.3

Apply the distributive property.

$z = \frac{3 \cdot - 1 + 3 i - 2 i \cdot - 1 - 2 i i}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2

Simplify and combine like terms.

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Step 5.3.3.2.2.1

Simplify each term.

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Step 5.3.3.2.2.1.1

Multiply $3$ by $- 1$ .

$z = \frac{- 3 + 3 i - 2 i \cdot - 1 - 2 i i}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.1.2

Multiply $- 1$ by $- 2$ .

$z = \frac{- 3 + 3 i + 2 i - 2 i i}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.1.3

Multiply $- 2 i i$ .

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Step 5.3.3.2.2.1.3.1

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

$z = \frac{- 3 + 3 i + 2 i - 2 (i^{1} i)}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.1.3.2

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

$z = \frac{- 3 + 3 i + 2 i - 2 (i^{1} i^{1})}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.1.3.3

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

$z = \frac{- 3 + 3 i + 2 i - 2 i^{1 + 1}}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.1.3.4

Add $1$ and $1$ .

$z = \frac{- 3 + 3 i + 2 i - 2 i^{2}}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.1.4

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

$z = \frac{- 3 + 3 i + 2 i - 2 \cdot - 1}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.1.5

Multiply $- 2$ by $- 1$ .

$z = \frac{- 3 + 3 i + 2 i + 2}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.2

Add $- 3$ and $2$ .

$z = \frac{- 1 + 3 i + 2 i}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.2.2.3

Add $3 i$ and $2 i$ .

$z = \frac{- 1 + 5 i}{(- 1 - i) (- 1 + i)}$

Step 5.3.3.3

Simplify the denominator.

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Step 5.3.3.3.1

Expand $(- 1 - i) (- 1 + i)$ using the FOIL Method.

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Step 5.3.3.3.1.1

Apply the distributive property.

$z = \frac{- 1 + 5 i}{- 1 (- 1 + i) - i (- 1 + i)}$

Step 5.3.3.3.1.2

Apply the distributive property.

$z = \frac{- 1 + 5 i}{- 1 \cdot - 1 - 1 i - i (- 1 + i)}$

Step 5.3.3.3.1.3

Apply the distributive property.

$z = \frac{- 1 + 5 i}{- 1 \cdot - 1 - 1 i - i \cdot - 1 - i i}$

Step 5.3.3.3.2

Simplify.

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Step 5.3.3.3.2.1

Multiply $- 1$ by $- 1$ .

$z = \frac{- 1 + 5 i}{1 - 1 i - i \cdot - 1 - i i}$

Step 5.3.3.3.2.2

Multiply $- 1$ by $- 1$ .

$z = \frac{- 1 + 5 i}{1 - 1 i + 1 i - i i}$

Step 5.3.3.3.2.3

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

$z = \frac{- 1 + 5 i}{1 - 1 i + 1 i - (i^{1} i)}$

Step 5.3.3.3.2.4

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

$z = \frac{- 1 + 5 i}{1 - 1 i + 1 i - (i^{1} i^{1})}$

Step 5.3.3.3.2.5

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

$z = \frac{- 1 + 5 i}{1 - 1 i + 1 i - i^{1 + 1}}$

Step 5.3.3.3.2.6

Add $1$ and $1$ .

$z = \frac{- 1 + 5 i}{1 - 1 i + 1 i - i^{2}}$

Step 5.3.3.3.2.7

Add $- 1 i$ and $1 i$ .

$z = \frac{- 1 + 5 i}{1 + 0 - i^{2}}$

Step 5.3.3.3.2.8

Add $1$ and $0$ .

$z = \frac{- 1 + 5 i}{1 - i^{2}}$

Step 5.3.3.3.3

Simplify each term.

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Step 5.3.3.3.3.1

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

$z = \frac{- 1 + 5 i}{1 - - 1}$

Step 5.3.3.3.3.2

Multiply $- 1$ by $- 1$ .

$z = \frac{- 1 + 5 i}{1 + 1}$

Step 5.3.3.3.4

Add $1$ and $1$ .

$z = \frac{- 1 + 5 i}{2}$

Step 5.3.4

Split the fraction $\frac{- 1 + 5 i}{2}$ into two fractions.

$z = \frac{- 1}{2} + \frac{5 i}{2}$

Step 5.3.5

Move the negative in front of the fraction.

$z = - \frac{1}{2} + \frac{5 i}{2}$