x2 + p1x + q1 = 0

with p1, q1 not both 0. If the equation has two real roots, Harini uses them to create a new quadratic equation,

x2 + p2x + q2 = 0,

using the smaller of the two roots as p2 and the larger one as q2. For instance, if Harini's first equation was x2 + 2x − 3 = 0, which has roots −3 and 1, then her second equation would be x2 − 3x + 1 = 0. She keeps going in this way: at each step n, if the equation

x2 + pnx + qn = 0

has two real roots, Harini uses them as the coefficients of the next equation,

x2 + pn+1x + qn+1 = 0,

always with the smaller root as pn+1 and the larger root as qn+1. (A repeated root counts as two equal roots, in which case pn+1 = qn+1.) She stops when she gets to an equation that does not have real roots.

- Prove that this process cannot continue forever.
- What is the maximal possible length of Harini's sequence of equations?