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How can one solve polynomial of the form ax^3+bx+c equal zero e.g x^3-27x-27 equal zero


I don't understand the approach u used.make it more clearer by using signs that are understandable.don't use the dollar sign ($)

Introduce two variables $u$ and $v$ linked by the condition

$u + v = x$,

a straightforward substitution transfroms the initial equation to the following

$u^{3}+v^{3}+(3uv + b)(u+v)+ c =0$

Impose the additional condition

$3uv + b = 0$

to deduce the system

$u^3 + v^3 = - b $ and $u^3 v^3 = -\frac{b^3}{27}$

that is equivalent to the fact that the $u^3$ and $v^3$ are roots of the quadratic equation

$z^2 +c z - \frac{b^3}{27} = 0$.

The method is due to Scipione del Ferro and Tartaglia, published by Gerolamo Cardano in 1545.

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