Question

Answer the following:

Find the square root of 15 – 8i

Answer 1

Let `sqrt(15 - 8"i")` = x + yi, where x, y ∈ R

On squaring both sides, we get,

15 – 8i = (x + yi)² = x² + y²i² + 2xyi

∴ 15 – 8i = (x² – y²) + 2xyi     ...[∵ i² = – 1]

Equating the real and imaginary parts separately, we get,

x² – y² = 15 and 2xy = – 8

∴ y = `-4/x`

∴ `x^2 - (-4/x)^2` = 15

∴ `x^2 - 16/x^2` = 15

∴ x⁴ – 16 = 15x²

∴ x⁴ – 15x² – 16 = 0

∴ (x² – 16)(x² + 1) = 0

∴ x² = 16 or x² = – 1

Now x is a real number

∴ x² ≠ – 1

∴ x² = 16

∴ x = ± 4

When x = 4, y = `(-4)/4` = – 1

When x = – 4, y = `(-4)/-4` = 1

∴ the square roots of 15 – 8i are 4 – i and – 4 + i, i.e., ± (4 – i).