In the quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then, the equation p[p(x)]=0 has
Neither real nor purely imaginary roots
If a quadratic equation has purely imaginary roots, then the coefficient of x must be equal to zero.
Let p(x)=ax2+b with a,b of same sign and a,b ϵ R.
Then, p[p(x)]=a(ax2+b)2+b
p(x) has imaginary roots say ix.
Then, ax2+b ϵ R and (ax2+b)2>0
∴a(ax2+b)2+b≠0,∀ x ϵ R
Thus, p[p(x)]≠0,∀ x ϵ R.
Hence, p[p(x)]=0 has complex roots.