Question:
Simplify each of the following and express it in the form a + ib :
$(1-i)^{2}(1+i)-(3-4 i)^{2}$
Solution:
Given: $(1-i)^{2}(1+i)-(3-4 i)^{2}$
$=\left(1+i^{2}-2 i\right)(1+i)-\left(9+16 i^{2}-24 i\right)$
$\left[\because(a-b)^{2}=a^{2}+b^{2}-2 a b\right]$
$=(1-1-2 i)(1+i)-(9-16-24 i)\left[\because i^{2}=-1\right]$
$=(-2 i)(1+i)-(-7-24 i)$
Now, we open the brackets
$-2 i \times 1-2 i \times i+7+24 i$
$=-2 i-2 i^{2}+7+24 i$
$=-2(-1)+7+22 i\left[\because, i^{2}=-1\right]$
$=2+7+22 i$
$=9+22 i$
