(i) Let x+iy be a root of the quadratic equation $z^2 +pz+ 1 = 0$ , where p is a real number.
Show that $x^2 − y^2 + px + 1 = 0$ and $(2x + p)y = 0$.
Show further that either $p = −2x$ or $p = −(x^2 + 1)/x$ with $x\neq 0 $.

Hence show that the set of points in the Argand diagram that can (as p varies) represent
roots of the quadratic equation consists of the real axis with one point missing and a
circle. This set of points is called the root locus of the quadratic equation.

(ii) Obtain and sketch in the Argand diagram the root locus of the equation
$pz^2 + z + 1 = 0$ .

(iii) Obtain and sketch in the Argand diagram the root locus of the equation
$pz^2 + p^2z + 2 = 0$ .

I've solved parts (i) and (ii) and attempted to use the same method to solve (iii) where I obtained the equations
$px^2 -py^2 + p^2x +2 = 0$ and $py(2x + p) = 0$
solving the second equation is simple by letting the brackets $2x + p = 0$
However, I've been unable to manipulate the first equation using $y=0$.
Please could you let me know if I'm actually using the right method or there's something I'm missing
Thank you!