#### The set of values of a such that the equation 2x^2 -2(2a+1)x+a(a-1)=0 has alpha and beta satisfying alpha < a < beta

Solution:   Let  $f(x)=2x^{2}-2(2a+1)x+a(a-1)$

a lies  between the roots of given equation .

$\\ \\ \Rightarrow \hspace{1cm}2f(a)< 0\Rightarrow f(a)< 0\\ \\ \Rightarrow \hspace{1cm}2a^{2}-2(2a+1)a+a(a-1)< 0\\ \\ \Rightarrow \hspace{1cm}-a^{2}-3a< 0\Rightarrow a(a+3)> 0\\ \\ \Rightarrow \hspace{1cm}a\in (\infty,-3)\cup(0,\infty)$