If the ratio of the roots of the equations $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ is the same,then:

Let $\alpha$ and $\beta$ be two distinct roots of the equation $x^3 + 3x^2 - 1 = 0$. The equation which has $(\alpha \beta)$ as its root is equal to

Let $f(n)=A(-2)^n+B(-3)^n$ for all $A, B \in \mathbb{R}$ and $n \in \mathbb{N}-\{1, 2\}$. If $f(n)+a f(n-1)+b f(n-2)=0$, then $(a+b)(b-a)=$

If $p$ and $q$ are the roots of the equation $x^2 + pq = (p + 1)x$,then $q=$

Suppose that the equation $ax^2+bx+c=0$ has roots $\alpha$ and $\beta$,both of which are different from $\frac{1}{3}$. Then,an equation whose roots are $\frac{1}{3\alpha-1}$ and $\frac{1}{3\beta-1}$ is

For the equation $3x^2 + px + 3 = 0, p > 0$,if one of the roots is the square of the other,then $p$ is equal to: