Answer 1

We have:

\[x^2 - 4x + 7 = 0\]

\[ \Rightarrow x^2 - 4x + 4 + 3 = 0\]

\[ \Rightarrow x^2 - 2 \times x \times 2 + 2^2 - (\sqrt{3}i )^2 = 0\]

\[ \Rightarrow (x - 2 )^2 - (\sqrt{3}i )^2 = 0\]

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

\[\Rightarrow (x - 2 + \sqrt{3}i) = 0\]  or, \[(x - 2 - \sqrt{3}i) = 0\]

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

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