Question

1/(2+i)²

Solve this question....​

Answer 1

$\large\underline{\sf{Solution-}}$

Given complex number is

$\rm \longmapsto\:\dfrac{1}{ {(2 + i)}^{2} }$

$\rm \:  =  \: \dfrac{1}{4 + {i}^{2} + 2 \times 2 \times i}$

$\rm \:  =  \: \dfrac{1}{4 - 1 + 4i}$

$\rm \:  =  \: \dfrac{1}{3+ 4i}$

$\rm \:  =  \: \dfrac{1}{3+ 4i} \times \dfrac{3 - 4i}{3 - 4i}$

$\rm \:  =  \: \dfrac{3 - 4i}{ {3}^{2} - {(4i)}^{2} }$

$\rm \:  =  \: \dfrac{3 - 4i}{9 -16 {i}^{2} }$

We know,

$\rm \longmapsto\:\tt{ {i}^{2} \: = \: - \: 1 \: }$

So, using this, we get

$\rm \:  =  \: \dfrac{3 - 4i}{9 -16( - 1)}$

$\rm \:  =  \: \dfrac{3 - 4i}{9 + 16}$

$\rm \:  =  \: \dfrac{3 - 4i}{25}$

$\rm \:  =  \: \dfrac{3}{25} - \dfrac{4}{25}i$