Examples

\frac{3 + i}{- 4 + 2 i}

$\frac{3 + i}{- 4 + 2 i}$

Step 1

Multiply the numerator and denominator of

\frac{3 + i}{- 4 + 2 i}

$\frac{3 + i}{- 4 + 2 i}$ by the conjugate of

- 4 + 2 i

$- 4 + 2 i$ to make the denominator real.

\frac{3 + i}{- 4 + 2 i} \cdot \frac{- 4 - 2 i}{- 4 - 2 i}

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

Step 2

Multiply.

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Step 2.1

Combine.

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(3 + i) (- 4 - 2 i)

$(3 + i) (- 4 - 2 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

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

Step 2.2.1.2

Apply the distributive property.

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

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

Step 2.2.1.3

Apply the distributive property.

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

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

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

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply

3

$3$ by

- 4

$- 4$ .

\frac{- 12 + 3 (- 2 i) + i \cdot - 4 + i (- 2 i)}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 + 3 (- 2 i) + i \cdot - 4 + i (- 2 i)}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.1.2

Multiply

- 2

$- 2$ by

3

$3$ .

\frac{- 12 - 6 i + i \cdot - 4 + i (- 2 i)}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i + i \cdot - 4 + i (- 2 i)}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.1.3

Move

- 4

$- 4$ to the left of

i

$i$ .

\frac{- 12 - 6 i - 4 \cdot i + i (- 2 i)}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 \cdot i + i (- 2 i)}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.1.4

Multiply

i (- 2 i)

$i (- 2 i)$ .

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Step 2.2.2.1.4.1

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 12 - 6 i - 4 i - 2 (i^{1} i)}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 i - 2 (i^{1} i)}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 12 - 6 i - 4 i - 2 (i^{1} i^{1})}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 i - 2 (i^{1} i^{1})}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.1.4.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{- 12 - 6 i - 4 i - 2 i^{1 + 1}}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 i - 2 i^{1 + 1}}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.1.4.4

Add

1

$1$ and

1

$1$ .

\frac{- 12 - 6 i - 4 i - 2 i^{2}}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 i - 2 i^{2}}{(- 4 + 2 i) (- 4 - 2 i)}$

\frac{- 12 - 6 i - 4 i - 2 i^{2}}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 i - 2 i^{2}}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 12 - 6 i - 4 i - 2 \cdot - 1}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 i - 2 \cdot - 1}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.1.6

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

\frac{- 12 - 6 i - 4 i + 2}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 i + 2}{(- 4 + 2 i) (- 4 - 2 i)}$

\frac{- 12 - 6 i - 4 i + 2}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 12 - 6 i - 4 i + 2}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.2

Add

- 12

$- 12$ and

2

$2$ .

\frac{- 10 - 6 i - 4 i}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 10 - 6 i - 4 i}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.2.2.3

Subtract

4 i

$4 i$ from

- 6 i

$- 6 i$ .

\frac{- 10 - 10 i}{(- 4 + 2 i) (- 4 - 2 i)}

$\frac{- 10 - 10 i}{(- 4 + 2 i) (- 4 - 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

$(- 4 + 2 i) (- 4 - 2 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 10 - 10 i}{- 4 (- 4 - 2 i) + 2 i (- 4 - 2 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 10 - 10 i}{- 4 \cdot - 4 - 4 (- 2 i) + 2 i (- 4 - 2 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 10 - 10 i}{- 4 \cdot - 4 - 4 (- 2 i) + 2 i \cdot - 4 + 2 i (- 2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$- 4$ by

$- 4$ .

$\frac{- 10 - 10 i}{16 - 4 (- 2 i) + 2 i \cdot - 4 + 2 i (- 2 i)}$

Step 2.3.2.2

Multiply

$- 2$ by

$- 4$ .

$\frac{- 10 - 10 i}{16 + 8 i + 2 i \cdot - 4 + 2 i (- 2 i)}$

Step 2.3.2.3

Multiply

$- 4$ by

$2$ .

$\frac{- 10 - 10 i}{16 + 8 i - 8 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply

$- 2$ by

$2$ .

$\frac{- 10 - 10 i}{16 + 8 i - 8 i - 4 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{- 10 - 10 i}{16 + 8 i - 8 i - 4 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{- 10 - 10 i}{16 + 8 i - 8 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 10 - 10 i}{16 + 8 i - 8 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{- 10 - 10 i}{16 + 8 i - 8 i - 4 i^{2}}$

Step 2.3.2.9

Subtract

$8 i$ from

$8 i$ .

$\frac{- 10 - 10 i}{16 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add

$16$ and

$0$ .

$\frac{- 10 - 10 i}{16 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{- 10 - 10 i}{16 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 4$ by

$- 1$ .

$\frac{- 10 - 10 i}{16 + 4}$

Step 2.3.4

Add

$16$ and

$4$ .

$\frac{- 10 - 10 i}{20}$

Step 3

Cancel the common factor of

$- 10 - 10 i$ and

$20$ .

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Step 3.1

Factor

$10$ out of

$- 10$ .

$\frac{10 (- 1) - 10 i}{20}$

Step 3.2

Factor

$10$ out of

$- 10 i$ .

$\frac{10 (- 1) + 10 (- i)}{20}$

Step 3.3

Factor

$10$ out of

$10 (- 1) + 10 (- i)$ .

$\frac{10 (- 1 - i)}{20}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

$10$ out of

$20$ .

$\frac{10 (- 1 - i)}{10 \cdot 2}$

Step 3.4.2

Cancel the common factor.

$\frac{10 (- 1 - i)}{10 \cdot 2}$

Step 3.4.3

Rewrite the expression.

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

Step 4

Split the fraction

$\frac{- 1 - i}{2}$ into two fractions.

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

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

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

Step 5.2

Move the negative in front of the fraction.

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

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