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

Consider the equation $x^2+4 x-n=0$, where $n \in[20,100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k, \frac{k}{2}\right)$ from the line $3 x+4 y+5=0$ is

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3} x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3 x-1=0$. If $P_n=$ $\alpha^n+\beta^n$ and $Q_n=\gamma^n+\hat{o}^n$, then $\frac{P_{25}+\sqrt{3} P_{24}}{2 P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to