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Let f(x) = x^2 + 2x. Find all real numbers x such that f(x) = f(f(x)).

 Sep 6, 2021
 #1
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x = -2   -1   0

 Sep 6, 2021
edited by ElectricPavlov  Sep 6, 2021
edited by ElectricPavlov  Sep 6, 2021
 #2
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x=-1 or x=-2

 Sep 6, 2021
 #3
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x=-1 or x=-2

 Sep 6, 2021
 #4
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Solution: x^2+2x=(x^2+2x)^2+2(x^2+2x)

 

               x^2+2x=x^4+2(x^2)(2x)+(2x)^2+2x^2+4x

               

               x^2+2x=x^4+4x^3+6x^2+4x

 

               0=x^4+4x^3+5x^2+2x

 

               Dividing everything by x gives: x^3+4x^2+5x+2=0

               Using the rational root theorem, you can check if any of these values make the equation equal to 0: 1, -1, 2, -2

               And -1 and -2 both check out.

 Sep 6, 2021

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