Advertisement Remove all ads

Find the square root of the following complex numbers: 7 + 24i - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Sum

Find the square root of the following complex numbers: 7 + 24i

Advertisement Remove all ads

Solution

Let `sqrt(7 + 24"i")` = a + bi, where a, b ∈ R
Squaring on both sides, we get
7 + 24i = (a + bi)2
∴ 7 + 24i = a2 + b2i2 + 2abi
∴ 7 + 24i = (a2 – b2) + 2abi         ...[∵ i2 = – 1]
Equating real and imaginary parts, we get
a2 – b2 = 7 and 2ab = 24

∴ a2 – b2 = 7 and b = `12/"a"`

∴ `"a"^2 - (12/"a")^2` = 7

∴ `"a"^2 - 144/"a"^2` = 7

∴ a4 – 144 = 7a2
∴ a4 – 7a2 – 144 = 0
∴ (a2 – 16)(a2 + 9) = 0
∴ a2 = 16 or a2 = – 9
But a ∈ R
∴ a2 ≠ – 9
∴ a2 = 16
∴ a = ± 4
When a = 4, b = `12/4` = 3

When a = – 4, b = `12/(-4)` = – 3

∴ `sqrt(7 + 24"i")` = ± (4 + 3i).

Concept: Square Root of a Complex Number
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board
Chapter 3 Complex Numbers
Exercise 3.2 | Q 1. (ii) | Page 40
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×