Roots and coefficients

Let p and q be real numbers such that $p\ne0$, $p^3\ne q$ and $p^3\ne-q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha+\beta=-p$ and $\alpha^3+\beta^3=q$, then a quadratic equation having $\dfrac\alpha\beta$ and $\dfrac\beta\alpha$ as its roots is

Let $a,b,c,p,q$ be real numbers. Suppose $\alpha,\beta$ are the roots of the equation $x^2+2px+q=0$ and $\alpha,\frac1\beta$ are the roots of the equation $ax^2+2bx+c=0$, where $\beta^2\notin\{-1,0,1\}$.

STATEMENT-1: $(p^2-q)(b^2-ac)\ge0$

and

STATEMENT-2: $b\ne pa$ or $c\ne qa$

Let $\alpha,\beta$ be the roots of the equation $x^2-px+r=0$ and $\frac\alpha2$, $2\beta$ be the roots of the equation $x^2-qx+r=0$. Then the value of $r$ is