Quadratic Equation question bank (Maths)

Consider the equation $\mathrm{x}^2+4 \mathrm{x}-\mathrm{n}=0$, where $\mathrm{n} \in[20,100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

If the set of all $\mathrm{a} \in \mathbf{R}$, for which the equation $2 x^2+(a-5) x+15=3 \mathrm{a}$ has no real root, is the interval $(\alpha, \beta)$, and $X=\{x \in Z: \alpha \lt x \lt \beta\}$, then $\sum_{x \in X} x^2$ is equal to :