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Consider this equation solving on WolframAlpha,

http://www.wolframalpha.com/input/?i=+solve+%5B+0+%3D+x^4+-+6x^2+-+8xcos%28+%282pi+%29%2F5+%29+-+2cos%28+%284pi%29%2F5%29+-+1+%5D

But the same equation on sage gives the roots,

[x == -1/2sqrt(-2sqrt(5) + 10) - 1, x == 1/2sqrt(-2sqrt(5) + 10) - 1, x == -1/2sqrt(2sqrt(5) + 6) + 1, x == 1/2sqrt(2sqrt(5) + 6) + 1]

It seems that the first two roots given by WolframAlpha differ from the last two roots given by Sage.

Why?

Consider this equation solving on input to WolframAlpha,

http://www.wolframalpha.com/input/?i=+solve+%5B+0+%3D+x^4+-+6x^2+-+8solve [ 0 = x^4 - 6x^2 - 8xcos%28+%282pi+%29%2F5+%29+-+2cos%28+%284pi%29%2F5%29+-+1+%5Dcos( (2pi )/5 ) - 2cos( (4pi)/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,

h(x) = x^4 - 6x^2 - 8xcos( (2pi )/5 ) - 2cos( (4pi)/5) - 1 h(x).solve(x)

[x == -1/2sqrt(-2sqrt(5) + 10) - 1, x == 1/2sqrt(-2sqrt(5) + 10) - 1, x == -1/2sqrt(2sqrt(5) + 6) + 1, x == 1/2sqrt(2sqrt(5) + 6) + 1]

It seems that the first two roots given by WolframAlpha differ from the last two roots given by Sage.

Why?

Consider this input to WolframAlpha,

solve [ 0 = x^4 - 6x^2 - 8xcos( (2pi )/5 ) - 2cos( (4pi)/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,

h(x) = x^4 - 6x^2 - 8xcos( (2pi )/5 ) - 2cos( (4pi)/5) - 1 h(x).solve(x)

[x == -1/2sqrt(-2sqrt(5) + 10) - 1, x == 1/2sqrt(-2sqrt(5) + 10) - 1, x == -1/2sqrt(2sqrt(5) + 6) + 1, x == 1/2sqrt(2sqrt(5) + 6) + 1]

It seems that the first two roots given by WolframAlpha differ from the last two roots given by Sage.

Why?

Consider this input to WolframAlpha,

solve [ 0 = x^4 - 6x^2 - 8xcos( (2pi )/5 ) - 2cos( (4pi)/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,

h(x) = x^4 - 6x^2 - 8xcos( (2pi )/5 ) - 2cos( (4pi)/5) - 1

h(x).solve(x)

[x == -1/2sqrt(-2sqrt(5) + 10) - 1, x == 1/2sqrt(-2sqrt(5) + 10) - 1, x == -1/2sqrt(2sqrt(5) + 6) + 1, x == 1/2sqrt(2sqrt(5) + 6) + 1]

It seems that the first two roots given by WolframAlpha differ from the last two roots given by Sage.

Why?

Consider this input to WolframAlpha,

solve [ 0 = x^4 - 6x^2 6*x^2 - 8xcos( (2pi 8*x*cos( (2*pi )/5 ) - 2cos( (4pi)/5) 2*cos( (4*pi)/5) - 1 ]
]

{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}}
0.175571}}

sage: h(x) = x^4 - 6x^2 6*x^2 - 8xcos( (2pi 8*x*cos( (2*pi )/5 ) - 2cos( (4pi)/5) 2*cos( (4*pi)/5) - 1 
  
h(x).solve(x)
 
sage: h(x).solve(x)
[x == -1/2sqrt(-2sqrt(5) -1/2*sqrt(-2*sqrt(5) + 10) - 1, x == 1/2sqrt(-2sqrt(5) 1/2*sqrt(-2*sqrt(5) + 10) -

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, - 1, x == -1/2sqrt(2sqrt(5) -1/2*sqrt(2*sqrt(5) + 6) + 1, x == 1/2sqrt(2sqrt(5) 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?