Practice section 8 of 9

Express $(3-i)^2$ in the form $a+ib$. Hence or otherwise find the roots of $(z+i)^2=-8+6i$. [4]

A graphic calculator is not to be used in answering this question.

Express $(1-4i)^2$ in the form $a+bi$. Hence find the roots of $\left(\dfrac z2+3\right)^2=15+8i$. [4]

Do not use a graphic calculator in answering this question.

(a) The diagram shows $y=f(x)$ for a cubic polynomial with real coefficients. What can be said about all roots of $f(x)=0$? [2]

(b) The equation $2z^2-(7+6i)z+11+ic=0$, where $c$ is a nonzero real number, has a root $z=3+4i$. Show that $c=-2$ and determine the other root in Cartesian form. Hence find the roots of $2w^2+(-6+7i)w-11+2i=0$. [6]

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

(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]

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

Find the roots of $iz^2-(5+i)z+2-6i=0$, giving answers in Cartesian form $a+bi$ where $a,b\in\mathbb R$. [2]

Without the use of a calculator, find the complex numbers $z,w$ satisfying $z-wi=3$ and $z^2-w+6+3i=0$. [6]