Determine the imaginary part of complex number z if z^2=3+4i .

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We have to determine the imaginary part of the complex number z for z^2 = 3 + 4i

Let z = x + yi

z^2 = x^2 + y^2*i^2 + 2xyi = 3+ 4i

=> x^2 - y^2 + 2xyi = 3 + 4i

equate the real and imaginary coefficients

we get x^2 - y^2 = 3 and 2xy = 4 or xy = 2 or x = 2/y

substitute x = 2/y in x^2 - y^2 = 3

=> 4/y^2 - y^2 = 3

=> 4 - y^4 = 3y^2

=> y^4 + 3y^2 - 4 = 0

=> y^4 + 4y^2 - y^2 - 4= 0

=> y^2( y^2 + 4) -1 ( y^2 + 4) = 0

=> (y^2 - 1)(y^2 + 4) = 0

=> y^2 = 1 and y^2 = -4

but y is a real coefficient, so we ignore y^2 = -4

y^2 = 1

=> y = 1

Therefore the imaginary part of the complex number z is 1.

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