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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

t h a n k y o u s i r