(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the solutions of:

[tex]x^3-3x-2=0[/tex]

using the Cardano's method.

2. Relevant equations

3. 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?????

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# Homework Help: Reduced cubic equation

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