How can we apply Cardano′s method in 2x^3 +5x^2 +x+2 i get u and v are solution of t^2 −56t+6859=0 but i think it′s wrong pls help


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Question Number 147419 by Kunal12588 last updated on 20/Jul/21

$H o w c a n w e a p p l y {C a r d a n o}^{'} s m e t h o d i n$ $2 x^{3} + 5 x^{2} + x + 2$ $i g e t u a n d v a r e s o l u t i o n o f t^{2} - 56 t + 6859 = 0$ $b u t i t h i n k {i t}^{'} s w r o n g p l s h e l p$
	Answered by mr W last updated on 20/Jul/21

	$x^{3} + \frac{5 x^{2}}{2} + \frac{x}{2} + 1 = 0$ $l e t x = s - \frac{5}{6}$ $s^{3} - 3 \times \frac{5}{6} s^{2} + 3 \times \frac{5^{2}}{6^{2}} s - \frac{5^{3}}{6^{3}} + \frac{5}{2} s^{2} - \frac{5}{2} \times 2 \times \frac{5}{6} s + \frac{5}{2} \times \frac{5^{2}}{6^{2}} + \frac{s}{2} - \frac{5}{2 \times 6} + 1 = 0$ $s^{3} - \frac{19}{12} s + \frac{47}{27} = 0$ $\sqrt{Δ} = \sqrt{{(- \frac{19}{36})}^{3} + {(\frac{47}{54})}^{2}} = \frac{\sqrt{3165}}{72}$ $s = \sqrt[3]{\frac{\sqrt{3165}}{72} - \frac{47}{54}} - \sqrt[3]{\frac{\sqrt{3165}}{72} + \frac{47}{54}}$ $\Rightarrow x = \sqrt[3]{\frac{\sqrt{3165}}{72} - \frac{47}{54}} - \sqrt[3]{\frac{\sqrt{3165}}{72} + \frac{47}{54}} - \frac{5}{6}$ $\approx - 2.416896$
	Commented by Kunal12588 last updated on 21/Jul/21

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