# [Maxima] Real solutions

Aleksas Domarkas aleksasd873 at gmail.com
Wed Apr 11 14:57:10 CDT 2012

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Theorem. Real roots of 8*x^3-6*x-1=0 is
[cos(%pi/9),cos((5*%pi)/9),cos((7*%pi)/9)]

Proof.
(%i1)  solve(8*x^3-6*x-1,x)\$
(%i2) sol:map(rhs,%)\$
(%i3) makelist(map(polarform,sol[k]),k,1,3);
(%o3)
[%e^((5*%i*%pi)/9)/2+%e^(-(5*%i*%pi)/9)/2,%e^((7*%i*%pi)/9)/2+%e^(-(7*%i*%pi)/9)/2,%e^((%i*%pi)/9)/2+%e^(-(%i*%pi)/9)/2]
(%i4) rectform(%);
(%o4) [cos((5*%pi)/9),cos((7*%pi)/9),cos(%pi/9)]
(%i5) sort(%);
(%o5) [cos(%pi/9),cos((5*%pi)/9),cos((7*%pi)/9)]

Test:
(%i6) float(%)\$ sort(%);
(%o7) [-0.766044443118978,-0.17364817766693,0.939692620785908]
(%i8) allroots(8*x^3-6*x-1)\$ sort(%);
(%o9) [x=-0.766044443118978,x=-0.17364817766693,x=0.939692620785908]

Aleksas D
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