An algebra problem by Ishan Ambekar
Algebra
Level
4
Let ax^2+bx+c=f(x).Given f(x)=x has no real roots, find the number of real roots of f(f(x))=x.
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1 solution
f(x)=x has complex roots simply means that f(x) does not intersect the line y=x.Since geometry does not depend on rotation,we rotate the coordinate axes by 45 degrees.We denote the rotated parabola by g(x).So now the question is reduced to- g(x)=0 has complex roots,find the number of real roots of g(g(x))=0.Now g(x)=0 has complex roots implies that for all real x ,g(x) is non zero.Since g(x) is real for all real x,g(g(x)) is non zero for all real x.Therefore all roots of g(g(x)) are complex.