Question

Express the given equation in the form of a + ib, a, b ∈ R $i=\sqrt{-1}$. State the values of a and b.
$\frac{3+2i}{2-5i}+\frac{3-2i}{2+5i}$

Answer 1

To express the given complex number in the form of a+ib,
let add both the numbers

$\frac{3+2i}{2-5i}+\frac{3-2i}{2+5i} \\ \\ = \frac{(3 + 2i)(2 + 5i) + (3 - 2i)(2 - 5i)}{(2 - 5i)(2 + 5i)} \\ \\ = \frac{6 + 15i + 4i + 10 {i}^{2} + 6 - 15i - 4i + 10 {i}^{2} }{( {2)}^{2} - ( {5i)}^{2} } \\ \\ = \frac{12 + 20 {i}^{2} }{4 + 25} \\ \\ = \frac{12 - 20}{29} \\ \\ = \frac{ - 8}{29} \\ \\ so \\ \\ a + ib = \frac{ - 8}{29} + 0i \\ \\ a = \frac{ - 8}{29} \\ \\ b = 0$
Hope it helps you.