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 \(x\).
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. \(\displaystyle x= \frac{-1\pm\sqrt{1^2- 4(1)(1)}}{2(1)}= \frac{-1\pm\sqrt{-3}}{2}= \frac{1}{2}\pm \frac{\sqrt{3}}{2}i\).
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