Mathematics - Quadratic Equation Question with Solution | TestHub

Both the roots are positive
$\begin{array}{rl}& D\ge 0\\ & 4(a-3{)}^2-4\times 9(1-a)\ge 0\\ & a^2-6a+9-9+9a\ge 0\\ & a^2+3a\ge 0\\ & a(a+3)\ge 0\end{array}$
$\mathrm{a}\in (-\infty ,-3]\cup [0,\infty )$ ...(i)
$\begin{array}{rl}& -\frac{\mathrm{b}}{2\mathrm{a}}>0\\ & \frac{2(\mathrm{a}-3)}{2(\mathrm{a}-1)}>0\end{array}$
$a\in (-\infty ,1)\cup (3,\infty )$ ...(ii)
$f(0)=9>0$
Equation (i) $\cap$ (ii)
$\begin{array}{rl}& a\in (-\infty ,-3]\cup [0,1)\\ & 2\alpha +\beta +\gamma -6+0+1=7\end{array}$