The value of $a$ $(a \ge 3)$ for which the sum of the cubes of the roots of $x^2 - (a - 2)x + (a - 3) = 0$ assumes the least value is:

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3x^2-7x+5=0$,then the value of $\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}$ is

If one root of the equation $ax^2 + bx + c = 0$ is $n$ times the other root,then:

If $\alpha_1, \alpha_2$ are the roots of $x^2+ax+1=0$ and $\alpha_3, \alpha_4$ are the roots of $x^2+bx+1=0$,then $(\alpha_1+\alpha_3)(\alpha_2+\alpha_3)(\alpha_1+\alpha_4)(\alpha_2+\alpha_4) = $

The harmonic mean of two numbers is $-\frac{8}{5}$ and their geometric mean is $2$. The quadratic equation whose roots are twice those numbers is

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$ ($a \neq 0$; $a, b, c \in \mathbb{R}$),then $(1 + \alpha + \alpha^2)(1 + \beta + \beta^2)$ is . . . .