Simplify (2-3i)/(2+i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 3$ .

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

Multiply $- 3 i (- i)$ .

Multiply $- 1$ by $- 3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $3$ by $- 1$ .

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

Subtract $3$ from $4$ .

$\frac{1 - 2 i - 6 i}{(2 + 1 i) (2 - i)}$

Subtract $6 i$ from $- 2 i$ .

$\frac{1 - 8 i}{(2 + 1 i) (2 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{1 - 8 i}{2 (2 - i) + 1 i (2 - i)}$

Apply the distributive property.

$\frac{1 - 8 i}{2 \cdot 2 + 2 (- i) + 1 i (2 - i)}$

Apply the distributive property.

$\frac{1 - 8 i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{1 - 8 i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Multiply $- 1$ by $2$ .

$\frac{1 - 8 i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)}$

Multiply $2$ by $1$ .

$\frac{1 - 8 i}{4 - 2 i + 2 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{1 - 8 i}{4 - 2 i + 2 i - i i}$

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

$\frac{1 - 8 i}{4 - 2 i + 2 i - (i^{1} i)}$

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

$\frac{1 - 8 i}{4 - 2 i + 2 i - (i^{1} i^{1})}$

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

$\frac{1 - 8 i}{4 - 2 i + 2 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{1 - 8 i}{4 - 2 i + 2 i - i^{2}}$

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

$\frac{1 - 8 i}{4 + 0 - i^{2}}$

Add $4$ and $0$ .

$\frac{1 - 8 i}{4 - i^{2}}$

Simplify each term.

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

$\frac{1 - 8 i}{4 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{1 - 8 i}{4 + 1}$

Add $4$ and $1$ .

$\frac{1 - 8 i}{5}$

Step 3

Split the fraction $\frac{1 - 8 i}{5}$ into two fractions.

$\frac{1}{5} + \frac{- 8 i}{5}$

Step 4

Move the negative in front of the fraction.

$\frac{1}{5} - \frac{8 i}{5}$

Simplify (2-3i)/(2+i)

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