The quadratic expression $(2x+1)^{2} - px + q \neq 0$ for any real $x$, if

Let the set of all values of $p \in R$,for which both the roots of the equation $x^2-(p+2)x+(2p+9)=0$ are negative real numbers,be the interval $(\alpha, \beta]$. Then $\beta-2\alpha$ is equal to

For what minimum value of $k$ do both roots of the equation $x^2 - 8kx + 16(k^2 - k + 1) = 0$ exist,are real,distinct,and at least $4$?

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 - 3x + a = 0$,where $a \in R$ and $\alpha < 1 < \beta$,then which of the following is true?

The value of $k$ for which the equation $x^2 - 3x + k = 0$ has at least one real root in $[0, 1]$ is

If non-zero real numbers $p$ and $q$ exist such that $\min f(x) > \max g(x)$,where $f(x) = x^2 + 2px + 2q^2$ and $g(x) = -x^2 - 2qx + p^2$ for $x \in \mathbb{R}$,find the set of values containing $|\frac{2p}{q}|$.