Question

The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has

• A. one purely imaginary root
• B. all real roots
• C. two real and two purely imaginary roots
• D. neither real nor purely imaginary roots

Answer 1

The correct option is D neither real nor purely imaginary roots

Quadratic equation with real coefficients and purely imaginary roots can be considered as
$p\left(x\right)=x^2+a=0wherea>0andaϵRThep\left[p\right(x\left)\right]=0\Rightarrow (x^2+a{)}^2+a=0\Rightarrow x^4+2a{x}^2+(a^2+a)=0\Rightarrow x^2=\frac{-2a\pm \sqrt{4a^2-4a^2-4a}}{2}\Rightarrow x^2=-a\pm \sqrt{a}i\Rightarrow x=\sqrt{-a\pm \sqrt{a}i}=\alpha \pm i\beta where\alpha ,\beta \le 0$
$\therefore p\left[p\right(x\left)\right]=0$ has complex roots which are neither purely real nor purely imaginary.