How to solve $x^3=-1$?

$x^3+1=0\implies (x+1)(x^2-x+1)=0$

If $x+1=0,x=-1$

else $x^2-x+1=0$ then $x=\frac{1±\sqrt{1-4}}{2\cdot1}=\frac{1±\sqrt3i}{2}$ using the well known Quadratic Formula.

Alternatively, using Euler's identity and Euler's formula like other two solutions, for $x^n=-1=e^{i(2m+1)\pi}$ as $e^{i\pi}=-1$ where $m$ any any integer, $n$ is a natural number .

We know $n$-degree equation has exactly $n$ roots, so the roots of $x^n+1=0$ are $e^{\frac{i(2m+1)\pi}{n}}=\cos\frac{(2m+1)\pi}{n}+i\sin \frac{(2m+1)\pi}{n}$ where $m=0,1,2,...n-1$. It's just customary, not mandatory that we have defined $m$ to assume this range of values, in fact it can assume any $n$ in-congruent values (for example, consecutive values like $r,r+1,...,r+n-1,$ where r is any integer), the reason is explained below.

(1)Using Repeated Root Theorem, let $f(x)=x^n+R$ where $n>1$ and $R ≠ 0\implies x≠0$

So, $\frac{df}{dx}=nx^{n-1}, \frac{df}{dx}=0$ does not have any non-zero root.

Clearly, $f(x)=x^n+R$ can not have any repeated root unless $R=0$

(2)Let $e^{\frac{i(2a+1)\pi}{n}}=e^{\frac{i(2b+1)\pi}{n}}$ $\implies e^{\frac{2i(a-b)\pi}{n}}=1=e^{2k\pi i}$ where $k$ is any integer.

$\implies a-b=kn\implies a≡b\pmod n$, so any $n$ in-congruent values of $m$ will give us essentially the same set of n distinct solutions.

Here $n=3$ so, let's take $m=0,1,2$.

$m=0\implies \cos\frac{\pi}{3}+i\sin \frac{\pi}{3}=\frac{1+i\sqrt3}{2} $

$m=1\implies \cos \pi+i\sin\pi=-1 $

$m=2\implies \cos\frac{5\pi}{3}+i\sin \frac{5\pi}{3}=\frac{1-i\sqrt3}{2} $

Let $x=a+bi$, where $a$ and $b$ are real. Then $(a+bi)^3 = -1$. Expanding the left-hand side gives $$a^3 +3a^2bi -3ab^2 -b^3i = -1.$$

We can separate the real and imaginary parts of this equation: $$\begin{align} a^3 -3ab^2 & = -1 \\ 3a^2b - b^3 & = 0 \end{align}$$

Taking the obvious $b=0$ solution gives us $a=-1$ and thus the real solution

$$x =-1.$$

So suppose $b\ne 0$. Then the second equation has $3a^2 = b^2$, so $b = \pm a\sqrt3$. Putting $3a^2$ for $b^2$ in the first equation gives $$ a^3 - 9a^3 = -1$$ so $a = \frac12.$ Since $b = \pm a\sqrt3$, we have $b=\pm\frac{\sqrt3}2$. This gives us the other two solutions, namely

$$x=\frac12 \pm\!\frac{\sqrt3}2i.$$

Set $\displaystyle x=re^{i \theta}$. So $\displaystyle r^3e^{i3\theta}= x^3= -1= e^{i \pi}$, hence $r^3=1$ and $3 \theta= \pi [2\pi]$. Finally, $r=1$ and $\displaystyle \theta = \frac{\pi}{3} \left[\frac{2\pi}{3} \right]$ so you get three solutions: $$x_1= e^{i \pi/3}= \frac{1}{2}+ i \frac{\sqrt{3}}{2}, \ x_2=e^{-i 2 \pi /3}= \frac{1}{2} - i \frac{\sqrt{3}}{2}, \ x_3= e^{i \pi}=-1.$$