Simplify -2/(5-i)

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

Step 1

Multiply the numerator and denominator of $\frac{- 2}{5 - i}$ by the conjugate of $5 - i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 2$ by $5$ .

$\frac{- 10 - 2 i}{(5 - i) (5 + i)}$

Simplify the denominator.

Expand $(5 - i) (5 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 10 - 2 i}{5 (5 + i) - i (5 + i)}$

Apply the distributive property.

$\frac{- 10 - 2 i}{5 \cdot 5 + 5 i - i (5 + i)}$

Apply the distributive property.

$\frac{- 10 - 2 i}{5 \cdot 5 + 5 i - i \cdot 5 - i i}$

Simplify.

Multiply $5$ by $5$ .

$\frac{- 10 - 2 i}{25 + 5 i - i \cdot 5 - i i}$

Multiply $5$ by $- 1$ .

$\frac{- 10 - 2 i}{25 + 5 i - 5 i - i i}$

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

$\frac{- 10 - 2 i}{25 + 5 i - 5 i - (i^{1} i)}$

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

$\frac{- 10 - 2 i}{25 + 5 i - 5 i - (i^{1} i^{1})}$

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

$\frac{- 10 - 2 i}{25 + 5 i - 5 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 10 - 2 i}{25 + 5 i - 5 i - i^{2}}$

Subtract $5 i$ from $5 i$ .

$\frac{- 10 - 2 i}{25 + 0 - i^{2}}$

Add $25$ and $0$ .

$\frac{- 10 - 2 i}{25 - i^{2}}$

Simplify each term.

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

$\frac{- 10 - 2 i}{25 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{- 10 - 2 i}{25 + 1}$

Add $25$ and $1$ .

$\frac{- 10 - 2 i}{26}$

Step 3

Cancel the common factor of $- 10 - 2 i$ and $26$ .

Factor $2$ out of $- 10$ .

$\frac{2 (- 5) - 2 i}{26}$

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

$\frac{2 (- 5) + 2 (- i)}{26}$

Factor $2$ out of $2 (- 5) + 2 (- i)$ .

$\frac{2 (- 5 - i)}{26}$

Cancel the common factors.

Factor $2$ out of $26$ .

$\frac{2 (- 5 - i)}{2 \cdot 13}$

Cancel the common factor.

$\frac{2 (- 5 - i)}{2 \cdot 13}$

Rewrite the expression.

$\frac{- 5 - i}{13}$

Step 4

Split the fraction $\frac{- 5 - i}{13}$ into two fractions.

$\frac{- 5}{13} + \frac{- i}{13}$

Step 5

Simplify each term.

Move the negative in front of the fraction.

$- \frac{5}{13} + \frac{- i}{13}$

Move the negative in front of the fraction.

$- \frac{5}{13} - \frac{i}{13}$

Simplify -2/(5-i)

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