**Q Find the square root of the complex number 5 -12i.**

Let

⇒ 5 - 12 i = (x + iy)^{2}

⇒ 5 - 12 i = x^{2} - y^{2 }+ 2i xy

⇒ 5 - 12 i = (x^{2} - y^{2}) + 2i xy

⇒ x^{2} - y^{2 }= 5 ... (1)

and 2xy = -12

Now,

Solving equation (1) and (2),

*x*^{2} = 9 and *y*^{2} = 4

⇒ *x* = ± 3 and *y* = ± 2

From equation (1), 2*xy* is negative, so *x* and *y* are of opposite signs.

∴ *x* = 3, *y* = – 2 or *x* = – 3, *y* = 2

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