Question

Answer the following:

Find the square root of −16 + 30i

Answer 1

Let `sqrt(-16 + 30"i")` = a + bi, where a, b ∈ R

Squaring on both sides, we get

−16 + 30i = a² + b²i² + 2abi

∴ −16 + 30i = (a² − b²) + 2abi    ...[∵ i² = − 1]

Equating real and imaginary parts, we get

a² − b² = −16 and 2ab = 30

∴ a² − b² = −16 and b = `15/"a"`

∴ `"a"^2 - (15/"a")^2` = −16

∴ `"a"^2 - 225/"a"^2` = −16

∴ a⁴ − 225 = − 16a²

∴ a⁴ + 16a² − 225 = 0

∴ (a² + 25) (a² − 9) = 0

∴ a² = − 25 or a² = 9

But a ∈ R 

∴ a² ≠ −25

∴ a² = 9

∴ a = ± 3

When a = 3, b = `15/3` = 5

When a = − 3, b = `15/(-3)` = − 5

∴ `sqrt(-16 + 30"i")` = ± (3 + 5i)