Keeping p constant and changing q the two real roots move along the real axis to meet where the roots coincide when p² = 4q. The pair of conjugate complex roots

-p ±   _____ Öp2-4q   2

then move along the vertical line in the Argand diagram through Re[z] = -p/2.

(2)(a) x² - 10x +16 = 0 has solutions 2and 8.
(b)x² - 8x + 16 = 0 has coincident solutions 4and 4.
(c) x² - 6x + 16 = 0 has complex conjugate roots 3±7i.

Keeping q constant and changing p, as the product of the roots is constant, one real root moves away from the origin and the other moves from infinity towards the origin until they coincide. The pair of complex conjugate roots then move along two semicircular arcs until they meet again on the real axis. The root which started at the origin becomes a real root moving off to infinity while the root which started at infinity moves in towards the origin.

(3) The roots are real when the discriminant p²-4q is positive, they coincide when the discriminant is zero and they are complex when the discriminant is negative. Hence, in the p-q plane, the parabola p² = 4q divides the plane into 2 regions. Above the parabola the points (p,q) are associated with complex roots, on this parabola with coincident roots and below this parabola with real roots.