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

First, we'll have to determine the complex
number.

We'll write the rectangular form of a complex
number:

z = a + bi

We'll raise
to square both sides:

z^2 =
(a+bi)^2

z^2 = a^2 + 2abi + b^2*i^2, but i^2 
=-1

z^2 = a^2 + 2abi -
b^2

From enunciation, we know
that:

z^2 = 3+4i

Comparing,
we'll get:

a^2 + 2abi - b^2 = 3 +
4i

a^2 - b^2 = 3 (1)

2ab =
4

ab = 2

b = 2/a
(2)

We'll substitute (2) in
(1):

a^2 - 4/a^2 = 3

We'll
multiply by a^2 all over:

a^4 - 3a^2 - 4 =
0

We'll substitute a^2 = t

t^2
- 3t - 4 = 0

We'll apply quadratic
formula:

t1 = [3 + sqrt(9 +
16)]/2

t1 = (3+5)/2

t1 =
4

a^2 = 4

a1 = 2 and a2 =
-2

b1 = 2/a1 = 2/2 = 1

b2 =
-1

t2 = (3-5)/2

t2 =
-1

a^2 = -1

a3 = i and a4 =
-i

b3 = 1/i = i/i^2 = -i

b4 =
i

The complex number are:

z1 =
2-i and z2=-2+i

z3= i - i^2

z3
= 1 + i

z4 = -i + i^2

z4 = -1
- i

The imaginary parts of z are: Im(z) = {-1
; 1}.