If x is and integer [x]=x and the equation
x3 + 2x2 = x3 + 2x2is always true.
If x is not an integer, let x=y+f where y is an integer and 0<f<1.
The equation becomes y3 + 2(y+f)2 = (y+f)3 + 2y2.
y^3 + 2y^2 + 4yf + 2f^2 = y^3 + 3y^2f + 3yf^2 + f^3 + 2y^2
4yf + 2f^2 = 3y^2f + 3yf^2 + f^3
4y + 2f = 3y^2 + 3yf + f^2
3y^2 + 3yf + f^2 - 4y -2f = 0
Which is a general conic. The discriminant test shows it is an ellipse, so there are not too many possibilities for y.
If y<0, f is non-real.
If y=0, f=0 or 2. Not allowed.
If y=1, f=-ϕ or ϕ-1. The second is a possibility around 0.618
If y=2, f=-2. Not allowed.
If y>2, f is non-real.
So the only non-integer x = y+f = ϕ.
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Posted by Jer
on 2024-04-16 13:37:17 |
Full solution
