Question

Express the given equation in the form of a + ib, a, b ∈ R $i=\sqrt{-1}$. State the values of a and b.
$\frac{i(4+3i)}{(1-i)}$

Answer 1

To solve this complex expression,first rationalised the denominator

$\frac{i(4+3i)}{(1-i)} \times \frac{1 + i}{1 + i} \\ \\since\: {i}^{2} =-1\\\\= \frac{(4i + 3 {i}^{2} )(1 + i)}{( {1)}^{2} - ( {i)}^{2} } \\ \\ = \frac{(4i - 3)(1 + i)}{ {1} + 1 } \\ \\ = \frac{4i + 4 {i}^{2} - 3 - 3i}{2} \\ \\ = \frac{ - 4 - 3 + i}{2} \\ \\ = \frac{ - 7 + i}{2} \\ \\ a + ib= \frac{ - 7}{2} + \frac{i}{2} \\ \\ here \: a = \frac{ - 7}{2} \\ \\ b = \frac{1}{2}$
Hope it helps you.