Usually the conjugate of a complex number `z` is denoted as `bar z .` using the eNotes site notation, one can write "bar z."
Now let's look at `z = x + i y` where `x` and `y` are real numbers, so `Re ( z ) = x` and `Im ( z ) = y .` Then, by the definition `bar z = x - i y` and `i z = i x - i^2 y = ix + y .`
This way, the equation becomes as follows:
`2 ( x + i y ) + 2 i ( x - i y ) = 2 ( x + i y + i x + y ) = 2 ( ( x + y ) + i ( x + y ) ) = 0 .`
For a complex number to be zero, both its real part and imaginary part must be zero. So we get `x + y = 0` and `x + y = 0 ,` actually only one condition (restriction). This means there are many such numbers `z :` fix any `x` and `y = -x ,` so the general solution is `x - i x,` `x in RR .`
This set of complex numbers may be shown on the complex plain as the bisector of the quadrants II and IV (a straight line).