# Quadratic Equation and Composite Function

I am trying to solve this but I think enough information is not given for example in (A) it will only hold true if we do not take x as imaginary (All 4 are given as correct in answer) . So can anyone help me with this question

Let quadratic equation p(x) = 0 (where p(x) = x^2 + bx + c) and equation p(p(p(x))) = 0 has a common root, then which of the following statement is/are correct.
(A) If b,c ∈ R, then b^2 – 4c ≥ 0.
(B) If P(0) = 1, then p(1) = 0.
(C) equations p(p(p(x))) = 0 and p(p(p(p(p(x))))) = 0 has at least two common root.
(D) zero is root of equation p(p(p(p(p(p(x)))))) = 0

Suppose $$x_0$$ is the common root of the equation $$p(x)=0$$ and $$p(p(p(x)))=0$$. Then we have $$p(p(p(x_0)))=p(p(0))=p(c)=0$$. This means that $$c$$ is a root of $$p(x)=0$$.
A) If $$b,c \in \mathbb{R}$$, then the quadratic will have a real root (namely, $$c$$), hence it will have both roots real, hence the discriminant is positive.
B) This is just if $$c=1$$. We know $$p(c)=0$$.
C) You can see that $$x_0$$ and $$c$$ are both common roots here.