Question

\[x^2 + x + 1 = 0\]

Answer 1

We have:

\[x^2 + x + 1 = 0\]

\[ \Rightarrow x^2 + x + \frac{1}{4} + \frac{3}{4} = 0\]

\[ \Rightarrow \left( x + \frac{1}{2} \right)^2 - \frac{3}{4} i^2 = 0\]

\[ \Rightarrow \left( x + \frac{1}{2} \right)^2 - \left( \frac{i\sqrt{3}}{2} \right)^2 = 0\]

\[ \Rightarrow \left( x + \frac{1}{2} + \frac{i\sqrt{3}}{2} \right) \left( x + \frac{1}{2} - \frac{i\sqrt{3}}{2} \right) = 0\]

\[\Rightarrow \left( x + \frac{1}{2} + \frac{i\sqrt{3}}{2} \right) = 0\] or \[\left( x + \frac{1}{2} - \frac{i\sqrt{3}}{2} \right) = 0\]

\[\Rightarrow x = - \frac{1}{2} - \frac{i\sqrt{3}}{2}\] or  \[x = - \frac{1}{2} + \frac{i\sqrt{3}}{2}\]

Hence, the roots of the equation are \[- \frac{1}{2} \pm \frac{i\sqrt{3}}{2} .\]