Question

find the roots of equation x^3-2x-5=0

Answer 1

$x^3-2x-5=0\\\\x^3-3 p\ x - 2\ q = 0,\ \ \ where\ p=2/3\ \ and\ \ q = 5/2\\\\Solution\ for\ x^3-3p\ x-2q =0\ is = x_k,\ \ k=0,1,2.\\\\ x_k=2\sqrt{p}*Cos\ [ \frac{1}{3} *\ Cos^{-1} (\frac{q}{p^{3/2}} ) - \frac{2\pi k}{3} \ ]$

$or,\\\\x= [q +\sqrt{q^2-p^3}]^{\frac{1}{3}}+[q-\sqrt{q^2-p^3}]^{\frac{1}{3}}\\\\ using\ this\ we\ get\ x=2.09455$