Question

Multiply the complex numbers: (1∕2 + 4i)^2

Options:

A)

15 3∕4 + 8i

B)

15 3∕4 + 4i

C)

–15 3∕4 + 4i

D)

–15 3∕4 + 8i

Answer 1

Answer:
`\large\boxed{-15 (3)/(4) +4i}`

Explanation:

`((1)/(2) +4i)^2 = ((1)/(2) +4i)((1)/(2) +4i)\\\\=(1)/(4) +2i+2i+16i^2\\\\=(1)/(4) +4i+16i^2\ \ \ \ \ note:\ \ i^2=-1\\\\=(1)/(4) +4i-16\\\\=\large\boxed{-15 (3)/(4) +4i}`

Answer 2

C

Explanation:

`\left(\frac 12 + 4i \right)^2\\\\=\left( \frac 12 \right)^2 + 2\cdot \frac 12 \cdot 4i+ (4i)^2\\ \\=\frac 14 + 4i + 16i^2\\\\=(1+16i+64i^2)/(4)\\\\=(1+16i-64)/(4)~~~~~~~~;[i^2 = -1]\\\\=\frac 14(16i-63)\\\\=-(63)/(4) + 4i\\\\=-15 \frac 34 +4i`