If $x$ is real, then the value of ${x^2} - 6x + 13$ will not be less than

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.

If the quadratic equation ${x^2} + \left( {2 - \tan \theta } \right)x - \left( {1 + \tan \theta } \right) = 0$ has $2$ integral roots, then sum of all possible values of $\theta $ in interval $(0, 2\pi )$ is $k\pi $, then $k$ equals 

The two roots of an equation ${x^3} - 9{x^2} + 14x + 24 = 0$ are in the ratio $3 : 2$. The roots will be

Two distinct polynomials $f(x)$ and $g(x)$ are defined as follows:

$f(x)=x^2+a x+2 ; g(x)=x^2+2 x+a$.If the equations $f(x)=0$ and $g(x)=0$ have a common root, then the sum of the roots of the equation $f(x)+g(x)=0$ is

The number of roots of the equation $|x{|^2} - 7|x| + 12 = 0$ is