complex numbers querie

[Sophdof1]

Hi,

Does anyone know how the author found the roots of z^2+z+1=0

 

[ksdhart2]

If I'm being pedantic and annoying, I don't specifically know how the author found the roots of this polynomial, but the most straightforward way I can think of it just to use the quadratic formula. For the purposes of that formula, it doesn't matter that the variable is $z$ rather than $x$.

 

[HallsofIvy]

Probably the quadratic formula! The roots of the equation $\displaystyle 4ax^2+ bx+ c= 0$ are $\displaystyle x= \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$. Here, the equation is $\displaystyle z^2+ z+ 1= 0$ so a= b= c= 1. $\displaystyle x= \frac{-1\pm\sqrt{1^2- 4(1)(1)}}{2(1)}= \frac{-1\pm\sqrt{-3}}{2}= \frac{1}{2}\pm \frac{\sqrt{3}}{2}i$.