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# Find the square root of: – 16 + 30i - Mathematics and Statistics

Sum

Find the square root of: – 16 + 30i

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#### Solution

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)

Concept: Square Root of a Complex Number
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board
Chapter 3 Complex Numbers
Miscellaneous Exercise 3 | Q 6. (i) | Page 43
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