Q. If $m$ is chosen in the quadratic equation $(m^2 + 1) x^2 - 3x + (m^2 + 1)^2 = 0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :
Solution:
$SOR = \frac{3}{m^{2} +1} \Rightarrow \left(S.O.R\right)_{max} = 3 $
when $m = 0$
$ \alpha+\beta=3 $
$ \alpha\beta = 1 $
$ \left|\alpha^{2} -\beta^{2}\right| =\left| \left|\alpha-\beta\right|\left(\alpha^{2} + \beta^{2} +\alpha\beta\right) \right| $
$ = \left|\sqrt{\left(\alpha-\beta\right)^{2}-\alpha\beta} \left(\left(\alpha+\beta\right)^{2} -\alpha\beta\right)\right| $
$ = \left|\sqrt{9-4} \left(9-1\right)\right| $
$= \sqrt{5} \times8 $
