Question

Find the square root of the following complex number:

7 + 24i 

Answer 1

Let `sqrt(7 + 24"i")` = x + yi, where x, y ∈ R.

On squaring both sides, we get,

7 + 24i = (x + yi)² = x² + y²i² + 2xyi

∴ 7+ 24i = (x² – y²) + 2xyi   ...[∵ i² = – 1]

Equating the real and imaginary parts separately, we get,

x² – y² = 7 and 2xy = 24

∴ y = `12/x`

∴ `x^2 - (12/x)^2` = 7

∴ `x^2 - 144/x^2` = 7

∴ x⁴ – 144 = 7x²

∴ x⁴ – 7x² – 144 = 0

∴ (x² – 16)(x² + 9) = 0

∴ x² = 16 or x² = – 9

Now x is a real number

∴ x² ≠ – 9

∴ x² = 16

∴ x = ± 4

When x = 4, y = `12/4` = 3

When x = – 4, y = `12/(-4)` = – 3

∴ the square roots of 7 + 24i are 4 + 3i and – 4 – 3i, i.e., ± (4 + 3i).