Find the all the roots of the polynomial x^{4}+ x^{2}+ 1.

The above is from an example in a text book. I attempted the question and solved it (I think) by

letting u = x^{2}

then solving the rewritten polynomial u^{2}+ u + 1 using the quadratic formula before solving for x.

I get the roots, $\displaystyle \sqrt[]{\frac {-1 + \sqrt[]{3} i} {2}}, -\sqrt[]{\frac {-1 + \sqrt[]{3} i} {2}}, \sqrt[]{\frac {-1 - \sqrt[]{3} i} {2}}, -\sqrt[]{\frac {-1 - \sqrt[]{3} i} {2}} $

But the book's author use completing the square to obtain the factors (x^{2}+ x + 1)(x^{2}- x + 1)

Which he then proceeds to solve using the quadratic formula to obtain $\displaystyle \frac {1 + \sqrt[]{3} i} {2}, \frac {-1 + \sqrt[]{3} i} {2}, \frac {1 - \sqrt[]{3} i} {2}, \frac {-1 - \sqrt[]{3} i} {2}$

My question is, why the different answers? Am I doing it wrong? That would be ironic because I learned that substitution trick from the very same book. Or is there something am missing?