If the equations $x^2+px+2=0$ and $x^2+x+2p=0$ have a common root,then the sum of the roots of the equation $x^2+2px+8=0$ is

Let $\lambda \neq 0$ be in $\mathbb{R}$. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-x+2\lambda=0$,and $\alpha$ and $\gamma$ are the roots of the equation $3x^{2}-10x+27\lambda=0$,then $\frac{\beta\gamma}{\lambda}$ is equal to:

If $x^2-3ax+14=0$ and $x^2+2ax-16=0$ have a common root,then $a^4+a^2=$

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant $G.P.$ such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root,then $\alpha(\beta + \gamma)$ is equal to

The equations $x^{2}+x+a=0$ and $x^{2}+ax+1=0$ have a common real root.

Let $a, b, c, p, q$ be real numbers. Suppose $\alpha, \beta$ are the roots of the equation $x^2+2px+q=0$ and $\alpha, \frac{1}{\beta}$ are the roots of the equation $ax^2+2bx+c=0$,where $\beta^2 \notin \{-1, 0, 1\}$.
$STATEMENT-1$: $(p^2-q)(b^2-ac) \geq 0$ and
$STATEMENT-2$: $b \neq pa$ or $c \neq qa$.