Solve for z, the equation |z|+ z = 2 + i

Z=a+ib

IzI + Z =2 + i

Ia+ibI + a+ib = 2+ i

(a^{2} + b^{2})^{1/2} + a+ib = 2+i

(a^{2} + b^{2})^{1/2} = 2 + i - a - ib

= (2-a) + (i-ib)

= (2-i) + i(1-b)

Comparing real and imaginary parts,

(a^{2} +b^{2})^{1/2} = 2-a 1-b=0

(a^{2} + 1)^{1/2} = 2-a b=1

a^{2} +1 = (2-a)^{2} = 4 + a^{2} - 4a

-4a = 1-4

a = 3/4