Question

Find the square root of -5 + 12 i

Answer 1

Answer:

[√-5+12 i] = ?

Let, √-5+12 i = x + iy

x=2 & y=3

The square root of -5+12 i = 2+3 i

Hope so these will be correct.

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Answer 2

Let the root of the given expression be

√(-5 + 12i) = √p - i√q

where p,q > 0

$\sqrt{ - 5 + 12i} = \sqrt{p} - i \sqrt{q} \\ squaring \: both \: sides \\ - 5 + 12i = p - q +i 2 \sqrt{pq} \\ comparing \: both \: sides \\ - 5 = p - q \: and \: 12 = 2 \sqrt{pq} \\ = >p = q - 5 \: and \: 36 = pq \\ = > 36 = (q - 5)q \: = > {q}^{2} - 5q - 36 = 0 \\ = > (q + 4)(q - 9) = 0 \: \\ = > q = - 4 \: and \: 9 \: ,but \: q > 0 \: , so \: q = 9\\ = > p = 4$

Therefore, √(-5+12i) = √4 - i√9

= 2 - 3i