For what set of values of $p$ do the roots of the equation $3x^2 + 2x + p(p - 1) = 0$ have opposite signs?

If roots of the equation $ax^2 + bx + c = 0$ are $(\alpha - \beta)$ and $(\gamma - \delta)$,and roots of the equation $Ax^2 + Bx + C = 0$ are $(\alpha + \delta)$ and $(\beta + \gamma)$,then $\left| \frac{a}{A} \right|$ is equal to (where $D_1$ and $D_2$ are discriminants of the given equations respectively).

The quadratic equation whose roots are the numbers having arithmetic mean $34$ and geometric mean $16$ is

Let $\alpha$ and $\beta$ be the roots of the equation $(x - a)(x - b) = c$,where $c \neq 0$. What are the roots of the equation $(x - \alpha)(x - \beta) + c = 0$?

The number of real solutions of the equation $|x|^2 - 3|x| + 2 = 0$ is

Which of the following is not a root of $f(x) = x^3 - 11x^2 + 36x - 36$?