# Algebra and quadratics equation sequence

#### MichNugget

Harini loves solving quadratic equations, but only if they have real roots. She starts with an equation
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.
1. Prove that this process cannot continue forever.
2. What is the maximal possible length of Harini's sequence of equations?

#### skipjack

Forum Staff
Where did you find this question?

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#### MichNugget

I found this question on Aops it was posted by a user and a similar question is there in my maths book that’s why I am interested to see the solution

#### skipjack

Forum Staff
Can you give the relevant AoPS link? Also, what maths book are you referring to and what was the similar question?

This problem is problem 6 in the current Mathcamp qualifying quiz, for which the instructions state that you may not consult or get help from anyone else on any aspect of the Qualifying Quiz.

#### EvanJ

Not that I need to know, but I'm wondering when the quiz is. After that it should be allowed to be posted here.

#### skipjack

Forum Staff
The deadline for applications this year is March 12. New questions are set each year.