Question

The equation of the smallest degree with real coefficients having 1 + i as one of the roots is

Options

• \[x^2 + x + 1 = 0\]

• \[x^2 - 2x + 2 = 0\]

• \[x^2 + 2x + 2 = 0\]

• \[x^2 + 2x - 2 = 0\]

Answer 1

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

We know that, imaginary roots of a quadratic equation occur in conjugate pair.
It is given that, 1 + i is one of the roots.
So, the other root will be  \[1 - i\] .

Thus, the quadratic equation having roots 1 + i and 1 - i is,

\[x^2 - \left( 1 + i + 1 - i \right)x + \left( 1 + i \right)\left( 1 - i \right) = 0\]

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