Question

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

Answer 1

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

\[ \Rightarrow x^2 - x + 2 = 0\]

\[ \Rightarrow x^2 - x + \frac{1}{4} + \frac{7}{4} = 0\]

\[ \Rightarrow x^2 - 2 \times x \times \frac{1}{2} + \left( \frac{1}{2} \right)^2 - \frac{7}{4} i^2 = 0\]

\[ \Rightarrow \left( x - \frac{1}{2} \right)^2 - \left( \frac{i\sqrt{7}}{2} \right)^2 = 0\]

\[ \Rightarrow \left( x - \frac{1}{2} + \frac{i\sqrt{7}}{2} \right) \left( x - \frac{1}{2} - \frac{i\sqrt{7}}{2} \right) = 0\]

\[\Rightarrow \left( x - \frac{1}{2} + \frac{\sqrt{7}}{2}i \right) = 0\]  or, \[\left( x - \frac{1}{2} - \frac{\sqrt{7}}{2}i \right) = 0\]

\[\Rightarrow x = \frac{1}{2} - \frac{\sqrt{7}}{2}i\] or, \[x = \frac{1}{2} + \frac{\sqrt{7}}{2}i\]

Hence, the roots of the equation are 

\[\frac{1}{2} \pm \frac{\sqrt{7}}{2}i\].