<s>[INST] Alternative Solution for Cubic Equation I was solving a problem and got to the following equation $x^3 + x^2 + x - 1 = 0  \; \; (1)$, numerically I found that the solution was:
$$x =\frac{1}{3} \left(-1 - \frac{2}{\left(17 + 3 \sqrt{33} \right)^{-1/3}} + \big(17 + 3 \sqrt{33} \big)^{-1/3} \right)$$
Which gives the same result as the professor. However, I'm suppose to write the solution of (1) as:
$$x = \frac{1}{3} \left(-1 -2\sqrt{2}\sinh\left( \frac{1}{3} \sinh^{-1} \left( \frac{17}{2\sqrt{2}}\right) \right)\right)$$
How I do get this expression?
 [/INST] I shall follow the steps given here.
We have $\Delta=-44$ so only one real root.
Using $p=\frac 23$, $q=-\frac{34}{27}$ and the hyperbolic method
$$t_0=\frac{2}{3} \sqrt{2} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{17}{2
   \sqrt{2}}\right)\right)$$
$$x=t_0-\frac b {3a}=\frac{1}{3} \left(-1+2 \sqrt{2} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{17}{2\sqrt{2}}\right)\right)\right)$$
I suppose that you have a sign error in what you wrote.
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