Basic Math Examples

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

Step 1

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

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

Step 2

Simplify the left side.

Tap for more steps...

Step 2.1

Cancel the common factor of $2 + i$ .

Tap for more steps...

Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Divide $z$ by $1$ .

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

Step 3

Simplify the right side.

Tap for more steps...

Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

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

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

Step 3.3

Multiply.

Tap for more steps...

Step 3.3.1

Combine.

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

Step 3.3.2

Simplify the numerator.

Tap for more steps...

Step 3.3.2.1

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

Tap for more steps...

Step 3.3.2.1.1

Apply the distributive property.

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

Step 3.3.2.1.2

Apply the distributive property.

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

Step 3.3.2.1.3

Apply the distributive property.

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

Step 3.3.2.2

Simplify and combine like terms.

Tap for more steps...

Step 3.3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.3.2.2.1.1

Multiply $3$ by $2$ .

$z = \frac{6 + 3 (- i) - 2 i \cdot 2 - 2 i (- i)}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.2

Multiply $- 1$ by $3$ .

$z = \frac{6 - 3 i - 2 i \cdot 2 - 2 i (- i)}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.3

Multiply $2$ by $- 2$ .

$z = \frac{6 - 3 i - 4 i - 2 i (- i)}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.4

Multiply $- 2 i (- i)$ .

Tap for more steps...

Step 3.3.2.2.1.4.1

Multiply $- 1$ by $- 2$ .

$z = \frac{6 - 3 i - 4 i + 2 i i}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.4.2

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

$z = \frac{6 - 3 i - 4 i + 2 (i^{1} i)}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.4.3

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

$z = \frac{6 - 3 i - 4 i + 2 (i^{1} i^{1})}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.4.4

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

$z = \frac{6 - 3 i - 4 i + 2 i^{1 + 1}}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.4.5

Add $1$ and $1$ .

$z = \frac{6 - 3 i - 4 i + 2 i^{2}}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.5

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

$z = \frac{6 - 3 i - 4 i + 2 \cdot - 1}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.1.6

Multiply $2$ by $- 1$ .

$z = \frac{6 - 3 i - 4 i - 2}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.2

Subtract $2$ from $6$ .

$z = \frac{4 - 3 i - 4 i}{(2 + 1 i) (2 - i)}$

Step 3.3.2.2.3

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

$z = \frac{4 - 7 i}{(2 + 1 i) (2 - i)}$

Step 3.3.3

Simplify the denominator.

Tap for more steps...

Step 3.3.3.1

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

Tap for more steps...

Step 3.3.3.1.1

Apply the distributive property.

$z = \frac{4 - 7 i}{2 (2 - i) + 1 i (2 - i)}$

Step 3.3.3.1.2

Apply the distributive property.

$z = \frac{4 - 7 i}{2 \cdot 2 + 2 (- i) + 1 i (2 - i)}$

Step 3.3.3.1.3

Apply the distributive property.

$z = \frac{4 - 7 i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 3.3.3.2

Simplify.

Tap for more steps...

Step 3.3.3.2.1

Multiply $2$ by $2$ .

$z = \frac{4 - 7 i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 3.3.3.2.2

Multiply $- 1$ by $2$ .

$z = \frac{4 - 7 i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)}$

Step 3.3.3.2.3

Multiply $2$ by $1$ .

$z = \frac{4 - 7 i}{4 - 2 i + 2 i + 1 i (- i)}$

Step 3.3.3.2.4

Multiply $- 1$ by $1$ .

$z = \frac{4 - 7 i}{4 - 2 i + 2 i - i i}$

Step 3.3.3.2.5

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

$z = \frac{4 - 7 i}{4 - 2 i + 2 i - (i^{1} i)}$

Step 3.3.3.2.6

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

$z = \frac{4 - 7 i}{4 - 2 i + 2 i - (i^{1} i^{1})}$

Step 3.3.3.2.7

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

$z = \frac{4 - 7 i}{4 - 2 i + 2 i - i^{1 + 1}}$

Step 3.3.3.2.8

Add $1$ and $1$ .

$z = \frac{4 - 7 i}{4 - 2 i + 2 i - i^{2}}$

Step 3.3.3.2.9

Add $- 2 i$ and $2 i$ .

$z = \frac{4 - 7 i}{4 + 0 - i^{2}}$

Step 3.3.3.2.10

Add $4$ and $0$ .

$z = \frac{4 - 7 i}{4 - i^{2}}$

Step 3.3.3.3

Simplify each term.

Tap for more steps...

Step 3.3.3.3.1

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

$z = \frac{4 - 7 i}{4 - - 1}$

Step 3.3.3.3.2

Multiply $- 1$ by $- 1$ .

$z = \frac{4 - 7 i}{4 + 1}$

Step 3.3.3.4

Add $4$ and $1$ .

$z = \frac{4 - 7 i}{5}$

Step 3.4

Split the fraction $\frac{4 - 7 i}{5}$ into two fractions.

$z = \frac{4}{5} + \frac{- 7 i}{5}$

Step 3.5

Move the negative in front of the fraction.

$z = \frac{4}{5} - \frac{7 i}{5}$