Distinguish a modulus square from a complex square

(a) The complex numbers $z,w$ satisfy $z+w^*+5i=10$ and $|w|^2=z+18+i$. Find $z$ and $w$.4 marks

(b)(i) Given that $2+i$ is a root of $z^2-5z+7+i=0$, find the second root in Cartesian form, showing your working.2 marks

(b)(ii) Hence find the roots of $-iw^2+5w+7i-1=0$.2 marks