Quadratic equation infinite sequence

MachNugget

New member
Joined
Feb 14, 2020
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2
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?
 

Romsek

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Joined
Nov 16, 2013
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784
Please show your current work and where you are having trouble.
 

Jomo

Elite Member
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Dec 30, 2014
Messages
5,407
Interesting question, specially part 2.
I am anxiously waiting to see your attempt.
Start off with an example. Maybe even use x^2 + 2x − 3 = 0 and see why it ends eventually.
 

Jomo

Elite Member
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Dec 30, 2014
Messages
5,407
Hint: if this was my problem I would think of using the quadratic formula repeatedly! I hope that helps!
 

Jomo

Elite Member
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Dec 30, 2014
Messages
5,407
Does anyone know a simple solution to this. I think that I can get somewhere doing what I am doing but it seems like there might be an easier route. If you prefer you can message me the solution. I will keep at it!
 
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