Find a solution for equation $x^3+x^2+x=1$.
I found $x=0.54369 (which is approximately equal to Cosine of 57 degree) by
trial and error as follows:
$(0.6)^3+(0.6)^2+0.6=1.176$...
$(0.5)^3+(0.5)^2+0.5=0.875$...
so $0.5< X < 0.6$
and $(0.54369)^3+(0.54369)^2+ 0.54369=1.000002936$
can anyone gives an anlytic solution for this equation?
Thanks for solution, it could be more useful if it was in detail.
If you use Cardano's method, you will notice that the equation has only a real root.
Using the depressed equation $$t^3+pt+q=0$$ you have $p=\frac{2}{3}$ and $q=-\frac{34}{27}$. Looking at the hyperbolic solution for one real root leads to $$t=\frac{2\sqrt{2}}{3} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{17}{2 \sqrt{2}}\right)\right)$$ Back to $x$ the solution write $$x=-\frac 13+\frac{2\sqrt{2}}{3} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{17}{2 \sqrt{2}}\right)\right)$$ which is $\approx 0.5436890$ while $\cos \left(\frac{57\pi }{180}\right)\approx 0.54463904$