2z + z' = 3 + 2i find lzl

The module of a complex number z = a + i*b is the positive
square root of the sum of the squares of the real part, a, and imaginary part,
b.

The conjugate of a complex number z is z' = a -
b*i

We'll calculate the sum from the left
side:

2z + z' = 2(a+b*i) + a -
b*i

We'll remove the
brackets:

2a + 2b*i + a -
b*i

We'll combine the real parts and the imaginary
parts:

2z + z' = 3a + b*i

The
real part and the imaginary part of the complex number from the left side have to be
equal with the real part and the imaginary part of the complex number form the right
side:

In our case, the complex number from the right
side is z = 3 + 2i 

3a + b*i = 3 +
2i 

We'll identify the real part and the imaginary
part:

Real part - Re(z) =
a:

3a = 3

We'll divide by
3:

a = 1

a =
3

Imaginary part - Im(z) = b

b
= 2

|z| = sqrt (a^2 + b^2)

|z|
= sqrt (1^2 + 2^2)

|z| = sqrt
(5)