[tex]\text{Use:}\ a^3-b^3=(a+b)(a^2-ab+b^2)[/tex]
[tex] x^3+1=0\\\\x^3+1^3=0\\\\(x+1)(x^2-(x)(1)+1^2)=0\\\\(x+1)(x^2-x+1)=0\iff x+1=0\ \vee\ x^2-x+1=0\\\\x+1=0\to x=-1\in\mathbb{R}-real\ solution\\\\x^2-x+1=0\\\\x^2-2\cdot x\cdot0.5+1=0\\\\\underbrace{x^2-2\cdot x\cdot0.5+0.5^2}_{(a-b)=a^2-2ab+b^2}-0.5^2+1=0\\\\(x-0.5)^2-0.25+1=0\\\\(x-0.5)^2+0.75=0\ \ \ |-0.75\\\\(x-0.5)^2=-0.75\iff x-0.5=\pm\sqrt{-0.75}\ \ \ |+0.5\\\\x=0.5-i\sqrt{0.75}\ \vee\ x=0.5+i\sqrt{0.75}\\\\x=\dfrac{1}{2}-i\sqrt{\dfrac{3}{4}}\ \vee\ x=\dfrac{1}{2}+i\sqrt{\dfrac{3}{4}} [/tex]
[tex]\boxed{x=\dfrac{1-i\sqrt3}{2}\ \vee\ x=\dfrac{1+i\sqrt3}{2}}[/tex]
[tex]\text{in standard form:}\ \boxed{x=\dfrac{1}{2}-\dfrac{\sqrt3}{2}i\ \vee\ x=\dfrac{1}{2}+\dfrac{\sqrt3}{2}i}[/tex]
