Evaluate (4-6i)/(-2+2i)

$\frac{4 - 6 i}{- 2 + 2 i}$

Step 1

Cancel the common factor of $4 - 6 i$ and $- 2 + 2 i$ .

Factor $2$ out of $4$ .

$\frac{2 \cdot 2 - 6 i}{- 2 + 2 i}$

Factor $2$ out of $- 6 i$ .

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

Factor $2$ out of $2 (2) + 2 (- 3 i)$ .

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

Cancel the common factors.

Factor $2$ out of $- 2$ .

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

Factor $2$ out of $2 i$ .

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

Factor $2$ out of $2 (- 1) + 2 (i)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

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

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

Step 3

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $- 1$ by $- 3$ .

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

Multiply $- 3 i (- i)$ .

Multiply $- 1$ by $- 3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $3$ by $- 1$ .

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

Subtract $3$ from $- 2$ .

$\frac{- 5 - 2 i + 3 i}{(- 1 + 1 i) (- 1 - i)}$

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

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $- 1$ by $- 1$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $i$ from $1 i$ .

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

Add $1$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

$\frac{- 5 + i}{2}$

Step 4

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

$\frac{- 5}{2} + \frac{i}{2}$

Step 5

Move the negative in front of the fraction.

$- \frac{5}{2} + \frac{i}{2}$

Evaluate (4-6i)/(-2+2i)

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