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Go Discriminant! (Posted on 2020-11-21) Difficulty: 3 of 5
Given 2 positive reals a and b. There exists 2 polynomials F(x)=x2+ax+b and G(x)=x2+bx+a such that all roots of polynomials F(G(x)) and G(F(x)) are real. Show that a and b are greater than 6.

No Solution Yet Submitted by Danish Ahmed Khan    
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Part way | Comment 1 of 3
F(G(x)) can only have real roots, if F(x) has real roots and 
G(F(x)) can only have real roots, if G(x) has real roots so:

a^2 >= 4b   and    b^2 >= 4a.  Combining:

a^4 >= (4b)^2 = 16b^2 >= 64a  so

a^3 >= 64   meaning   a >= 4

  Posted by FrankM on 2020-11-21 14:53:08
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