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I have the following cubic equation. Is it possible to solve it analytically in very "easy" way to a student only has a pre-calculus level:

The equation is:

$$x^3-3x+3=0$$

The real root I want to show is $x=-2.1038$

Update: Cardono's method is most suitable for my case so far, but I want to check if something is easier. Thank you all.

Aside note: I believe this cubic problem is unsuitable for students not exposed to numerical methods.

Blue
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M.K.A.B
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  • Do the well-known methods (you can find them in eg. wikipedia) not satisfy you? In what way are they unsatisfactory? You should explain this so that we can avoid telling you something you already know. – Trebor Dec 02 '23 at 16:41
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    Hint: for a cubic equation in the standard form $x^3+px=q$ the simplest way to proceed is to make Vieta's substitution, i.e. $x=w-\tfrac{p}{3w}$. – Anton Vrdoljak Dec 02 '23 at 16:44

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Using rational root theorem, it is easy to check that the equation has no rational root.

A good pre-calculus method is the bisection method. The given expression is $>0$ for $x=-2$ and $<0$ for $x=-3.$ So, we know there is a root between $-2$ and $-3$. Now simply apply the bisection method.

SARTHAK GUPTA
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