Maharashtra State BoardHSC Commerce 11th
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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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