## Loser66 one year ago How to simplify $$(2\overline{z}-i)(2z+i) =|2z+i|^2$$ Please, help

1. Loser66

nvm got it

2. anonymous

let $z=x+i y, \space \overline z=x-iy$ $\overline z \times z=x^2+y^2$ left hand side: $(2\overline{z}-i)(2z+i)=[4(x^2+y^2)+2i(x-iy)-2i(x+iy)+1]$ $[4(x^2+y^2)+2i(x-iy)-2i(x+iy)+1]=[(4x^2+4y^2)+2ix+2y-2ix+2y+1]$ $[4x^2+4y^2)+2ix+2y-2ix+2y+1]=[(4x^2+4y^2)+4y+1]$ right hand side: $|2z+i|^2=(\sqrt{(2x)^2+i(2y+1)^2}=4x^2+4y^2+4y+1$ so LHS=RHS