Number of roots of a complex equation

$|z|(z - 4 - i) + 2i = (5-i)z \Leftrightarrow z(|z| - 5 + i) = 4|z| + (|z| - 2)i$

Take the modulus of both sides, we get:

$|z|\sqrt{(|z|-5)^2+1} = \sqrt{(4|z|)^2+(|z| - 2)^2}$

Make the substitution $t=|z|$ , the above equation becomes:

$t\sqrt{(t-5)^2+1} = \sqrt{(4t)^2 = (t-2)^2} \Leftrightarrow (t - 1)(t^3 - 9t^2 + 4) = 0$

That cubic equation has 3 distinct roots $t\geq0$ , so there are 3 complex numbers $z$ that satisfies the equation