Consider the equation $x^2+x-n = 0$,where $n \in N$ and $n \in [5, 100]$. The total number of different values of $n$ such that the given equation has integral roots is:

The set of values of $x \in R$ satisfying the inequality $x^2 - 4x - 21 \leq 0$ is

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

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 number of values of $b$ for which there is an isosceles triangle with sides of lengths $b+5$,$3b-2$,and $6-b$ is

For which value of $a$ is the expression $x^2 - ax + 1 - 2a^2$ always positive for all real values of $x$?