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Physicsissuef
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Homework Statement
Find the solutions of:
[tex]x^3-3x-2=0[/tex]
using the Cardano's method.
Homework Equations
The Attempt at a Solution
[tex]x=u+v[/tex]
[tex](u+v)^3-3(u+v)-2=0[/tex]
[tex]u^3+v^3+(u+v)(3uv-3)-2=0[/tex]
[tex]3uv-3=0[/tex]
[tex]uv=1[/tex]
[tex]u^3v^3=1[/tex]
[tex]u^3+v^3=2[/tex]
[tex]u^3=v^3=1[/tex]
Now [tex]u=v=\sqrt[3]{1}[/tex].
I found u and v with [tex]z=\sqrt[3]{1}[/tex] (using complex numbers).
[tex]u=v=1[/tex]
[tex]u=v=\frac{-1}{2}+i\frac{sqrt{3}}{2}[/tex]
[tex]u=v=\frac{-1}{2}-i\frac{sqrt{3}}{2}[/tex]
x=u+v
[tex]x_1=2[/tex]
[tex]x_2=-1+i\sqrt{3}[/tex]
[tex]x_3=-1-i\sqrt{3}[/tex]
I substitute above and something is wrong. Where is the error?