Consider the quadratic equation ax² – bx + c = 0, a ,b , c $\in$ N. If the given equation has two distinct real roots belonging to (1, 2) then

$\mathrm{\alpha }-1,\mathrm{\beta }-1,2-\mathrm{\alpha },2-\mathrm{\beta }\in (0,1),\text{ use }$

$\mathrm{AM}\ge \mathrm{GM}\frac{(\mathrm{\alpha }-1)+(2-\mathrm{\alpha })}{2}\ge \sqrt{(\mathrm{\alpha }-1)(2-\mathrm{\alpha })}$

$\begin{array}{l}\Rightarrow (\mathrm{\alpha }-1)(2-\mathrm{\alpha })<\frac{1}{4},(\mathrm{\beta }-1)(2-\mathrm{\beta })<\frac{1}{4}\\ \Rightarrow 0<(\mathrm{\alpha }-1)(\mathrm{\beta }-1)(2-\mathrm{\alpha })(2-\mathrm{\beta })<\frac{1}{16}\\ \Rightarrow 0<\left(\frac{\mathrm{c}}{\mathrm{a}}-\frac{\mathrm{b}}{\mathrm{a}}+1\right)\left(4-\frac{2\mathrm{b}}{\mathrm{a}}+\frac{\mathrm{c}}{\mathrm{a}}\right)<\frac{1}{16}\\ 0<(\mathrm{a}-\mathrm{b}+\mathrm{c})(4\mathrm{a}-2\mathrm{b}+\mathrm{c})<\frac{{\mathrm{a}}^2}{16}\\ (\mathrm{a}-\mathrm{b}+\mathrm{c})(4\mathrm{a}-2\mathrm{b}+\mathrm{c})\ge 1\Rightarrow \frac{{\mathrm{a}}^2}{16}>1\Rightarrow \mathrm{a}\ge 5\end{array}$