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I need to analytically solve a cubic equation which has the form: $x^3 + bx^2 + c = 0.$ In this case, $b = 1-\frac{\sum_{i=1}^N x_i}{N}, c = \frac{\sum_{i=1}^N y_i}{N}.$ I am not looking for numerical methods. Thanks for any suggestions.

Kamal Saleh
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user996159
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    have you searched for cardano-tartaglia relationships? – Colver Jan 13 '23 at 14:23
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    to gain modern intuition, try https://www.wolframalpha.com/input?i=solve+a+x%5E3%2Bb+x%2Bc%3D0 . – janmarqz Jan 13 '23 at 14:24
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    @Colver is suggesting this technique. You'll need to set $y=x+b/3$ first. – J.G. Jan 13 '23 at 14:25
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    If you put $t=\frac 1x$ and divide through by $c (\neq 0)$ you will get a standard form in $t$ and any article or book which covers solutions of the cubic will give both method and formula. – Mark Bennet Jan 13 '23 at 14:25
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    Oopss https://www.wolframalpha.com/input?i=solve+a+x%5E3%2Bb+x%5E2%2Bc%3D0 . – janmarqz Jan 13 '23 at 14:25
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    You can follow my method (describes both quartic and cubic) in: https://math.stackexchange.com/questions/103988/solving-x4-10x3-21x2-40x-100-0/4601016#4601016 – Deepak Jan 13 '23 at 15:14

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