Embibe Experts Solutions for Chapter: Properties of Triangle, Exercise 2: Level 2

With usual notations, consider a $\Delta ABC$, with given $\angle A$ and side $a$. If $bc=x^2$, then such a triangle would exist, if ($x$ is a given positive real number)

The lengths of the sides of a triangle are $\alpha -\beta ,\alpha +\beta$ and $\sqrt{3{\alpha }^2+{\beta }^2},\left(\alpha >\beta >0\right)$. Its largest angle is  

The angles in a particular triangle are in arithmetic progression, and the side lengths are $4, 5, x.$ The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$, where $a, b$ and $c$ are positive integers. What is $a+b+c ?$