Question

Simplify each of the following and express it in the form a + ib :

(1 – i)² (1 + i) – (3 – 4i)²

Answer 1

Given: (1 – i)² (1 + i) – (3 – 4i)²

= (1 + i² – 2i)(1 + i) – (9 + 16i² – 24i)

[∵(a – b)² = a² + b² – 2ab]

= (1 – 1 – 2i)(1 + i) – (9 – 16 – 24i) [∵ i² = -1]

= (-2i)(1 + i) – (- 7 – 24i)

Now, we open the brackets

- 2i × 1 – 2i × i + 7 + 24i

= - 2i – 2i² + 7 + 24i

= - 2(-1) + 7 + 22i [∵, i² = -1]

= 2 + 7 + 22i

= 9 + 22i