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Question

Answer

$\frac{3}{2 - i} - \frac{2}{4 + i} = \frac{62 + 61 i}{85}$

Explanation

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$\frac{3}{2 - i} - \frac{2}{4 + i} \overset{◯}{1} \frac{5 i + 8}{- i ^{2} - 2 i + 8} \overset{◯}{2} \frac{5 i + 8}{1 - 2 i + 8} \overset{◯}{3} \frac{5 i + 8}{- 2 i + 9} \overset{◯}{4} \frac{62 + 61 i}{85}$
①	Subtract $\frac{2}{4 + i}$ from $\frac{3}{2 - i}$ to get $\frac{5 i + 8}{- i ^{2} - 2 i + 8}$ . To subtract raitonal expressions, both fractions must have the same denominator. We can create a common denominator by multiplying the first fraction by $i + 4$ and the second by $- i + 2$ . $\frac{3}{2 - i} - \frac{2}{4 + i} = \frac{3 \cdot ( i + 4 )}{( 2 - i ) \cdot ( i + 4 )} - \frac{2 \cdot ( - i + 2 )}{( 4 + i ) \cdot ( - i + 2 )} = = \frac{3 i + 12}{2 i + 8 - i ^{2} - 4 i} - \frac{- 2 i + 4}{2 i + 8 - i ^{2} - 4 i} = \frac{3 i + 12 - ( - 2 i + 4 )}{- i ^{2} - 2 i + 8} = = \frac{5 i + 8}{- i ^{2} - 2 i + 8}$
②	$- i^{2} = - (- 1) = 1$
③	Simplify denominator $1 - 2 i + 8 = - 2 i + 9$
④	Divide $8 + 5 i$ by $9 - 2 i$ to get $\frac{62 + 61 i}{85}$ . ( view steps )

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