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 exact set of values of $a$ for which the equation ${x^3}(x + 1) = 2(x + a)(x + 2a)$ has four real solutions is:

The number of solutions of the equation $\sqrt{3x^2+x+5} = x-3$ is

If $\alpha$ and $\beta$ are the roots of $x^2+3(a+3)x-9a=0$ such that the roots are equal for different values of $a$ (where $\alpha > \beta$ is not applicable as roots are equal,but let $\alpha$ be the root for $a=-9$ and $\beta$ be the root for $a=-1$),then the minimum value of the expression $x^2+\alpha x-\beta$ is:

The sum of the real roots of the equation $x^4-2x^3+x-380=0$ is

If $x$ is real,then the interval in which no value of the expression $\frac{2(x^2+2x-11)}{2x-5}$ lies,is