Why don't WolframAlpha's and Sage's answer match?
Consider this input to WolframAlpha,
solve [ 0 = x^4 - 6*x^2 - 8*x*cos( (2*pi )/5 ) - 2*cos( (4*pi)/5) - 1 ]
The solutions it gives are,
{x == (1 - Sqrt[5])/2 || x == (3 + Sqrt[5])/2 || x == (-2 - Sqrt[2 (5 - Sqrt[5])])/2 || x == (-2 + Sqrt[2 (5 - Sqrt[5])])/2, {-0.618034, 2.61803, -2.17557, 0.175571}}
But the same equation on sage gives the roots,
sage: h(x) = x^4 - 6*x^2 - 8*x*cos( (2*pi )/5 ) - 2*cos( (4*pi)/5) - 1
sage: h(x).solve(x)
[x == -1/2*sqrt(-2*sqrt(5) + 10) - 1, x == 1/2*sqrt(-2*sqrt(5) + 10) - 1,
x == -1/2*sqrt(2*sqrt(5) + 6) + 1, x == 1/2*sqrt(2*sqrt(5) + 6) + 1]
It seems that the first two roots given by WolframAlpha differ from the last two roots given by Sage.
Why?
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