Consider the quadratic equation $n{x}^2+7\sqrt{n} x+n=0$ , where $n$ is a positive integer. Which of the following statements are necessarily correct?

I. For any $n$, the roots are distinct.

II. There are infinitely many values of $n$ for which both roots are real.

III. The product of the roots is necessarily an integer.

Suppose $\mathrm{A},\mathrm{B},\mathrm{C}$ are defined as

$\mathrm{A}=a^{2}b+a{b}^2-a^{2}c-a{c}^2$

$\mathrm{B}=b^{2}c+b{c}^2-a^{2}b-a{b}^2$

$\mathrm{C}=a^{2}c+a{c}^2-b^{2}c-b{c}^2$

Where $a>b>c>0$ and the equation $A{x}^2+Bx+C=0$ has equal roots, then $a,b,c$ are in

Let $p\left(x\right)=x^2+ax+b$ have two distinct real roots, where $a,b$ are real number. Define $g\left(x\right)=p\left(x^3\right)$ for all real number $x$

Then, which of the following statements are true?
I. $g$ has exactly two distinct real roots.
II. $g$ can have more than two distinct real roots.
III. There exists a real number $\alpha$ such that $g\left(x\right)\ge \alpha$ for all real $x$