Question

How to use the discriminant to find out what type of solutions the equation has for `m^2 + m + 1 = 0`?

Answer 1

`m^2+m+1` is of the form `am^2+bm+c` with `a=b=c=1`.

The discriminant is given by the formula:

`Delta = b^2-4ac = 1^2 - (4xx1xx1) = 1-4 = -3`

`Delta < 0` so there are no real roots. There are two distinct complex roots.

As a matter of interest, the two roots are `omega` and `omega^2`, where

`omega = -1/2+sqrt(3)/2i`

is called the primitive cube root of unity.

Another way of expressing it is:

`omega = cos((2pi)/3)+isin((2pi)/3) = e^((2pi)/3i)`