Solve (1−4i1−1+i2)(5+i3−4i)

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Solve ( {{1}{{1 - 4i}} - {2}{{1 + i}}} )( {{{3 - 4i}}{{5 + i}}

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Solve $(\frac{1}{1 - 4 i} - \frac{2}{1 + i}) (\frac{3 - 4 i}{5 + i})$

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(\frac{1}{1 - 4 i} - \frac{2}{1 + i}) (\frac{3 - 4 i}{5 + i})

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\Rightarrow (\frac{1}{1 - 4 i} - \frac{2}{1 + i}) (\frac{3 - 4 i}{5 + i})

\Rightarrow [\frac{1 + 4 i}{( 1 - 4 i ) ( 1 + 4 i )} - \frac{2 ( 1 - i )}{( 1 + i ) ( 1 - i )}] [\frac{( 3 - 4 i ) ( 5 - i )}{( 5 + i ) ( 5 - i )}]

we know that

i^{2} = - 1 \times - 1 = - 1

so,

[\frac{1 + 4 i}{1 - ( 4 i ) ^{2}} - \frac{2 ( 1 - i )}{1 - i ^{2}}] [\frac{15 - 3 i - 20 i + 4 i ^{2}}{5 - i ^{2}}]

\displaystyle \Rightarrow \left [\frac{1+4i}{17}-\frac{1-i}{1}\right]\left[ \frac{11-23i}{6}\right]

\displaystyle \Rightarrow \left[\frac{1}{17}+\frac{4}{17}i-1+i\right]\left [\frac{11}{6}-\frac{23}{6}i\right]

\Rightarrow [\frac{- 16}{17} + \frac{21}{17} i] [\frac{11}{6} - \frac{23}{6} i]

\Rightarrow \frac{16}{17} \times \frac{11}{6} + \frac{16}{17} \times \frac{23}{6} i + \frac{21}{17} i \times \frac{11}{6} - \frac{21}{17} \times \frac{23}{6} (i^{2})

\Rightarrow \frac{- 88}{51} + \frac{184}{51} i + \frac{231}{102} i + \frac{483}{102}

\Rightarrow \frac{307}{102} + \frac{599}{102} i

\Rightarrow \frac{1}{102} [599 i + 307]

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