Question

The equation of the smallest degree with real coefficients having 1 + i as one of the roots is
(a) $x^2+x+1=0$
(b) $x^2-2x+2=0$
(c) $x^2+2x+2=0$
(d) $x^2+2x-2=0$

Answer 1

(b) $x^2-2x+2=0$

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,

$$\begin{gathered} x^2-\left(1+i+1-i\right)x+\left(1+i\right)\left(1-i\right)=0 \\ \Rightarrow x^2-2x+2=0 \end{gathered}$$