Question

$x$ satisfying inequality $ax^{2} + bx + c > 0$ is $(2,3)$

$$\begin{matrix} \Rightarrow & \begin{matrix} a < 0 & \\ \therefore & x^{2} + \frac{b}{a}x + \frac{c}{a} < 0 \\ & (x - 2)(x - 3) = x^{2} - 5x + 6 < 0 \end{matrix} & & \Rightarrow x^{2} + \frac{b}{a}x + \frac{c}{a} < 0 \\ & \frac{b}{a} = - 5,\frac{c}{a} = 6 & \Rightarrow & b = - 5a,c = 6a \\ \Rightarrow & c^{2} + bx + a < 0 & & \Rightarrow a\left( 6x^{2} - 5x + 1 \right) < 0 \\ \therefore & 6x^{2} - 5x + 1 > 0 & \Rightarrow & x \in \left( - \infty,\frac{1}{3} \right) \cup \left( \frac{1}{2},\infty \right) \end{matrix}$$