Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2-bx+c=0$ are distinct and both lie in the interval
(Brackets signify open interval, roots are $IN BETWEEN $ the numbers in each part.)
Find minimum possible value of $a, b, c.$
On my part, I solved for part 1, i.e. for distinct roots between (0,1). But for the next two parts, the things are getting a too bit messy.
While it may have similarity in question for given part 1 in stack exchange, there is no generalized method so that we can solve for other such intervals.
So please help, I am new to stack exchange.
For part 3, I tried by taking $0<m-2, 3-m, n-2, 3-n <1$ where m, n are the roots of the equation, and then using A. M. - G. M method, but i failed.
Please check another question of this type, But please don't provide with such answers as given in the link, as this is a question of an entrance exam, to be solved by hand, and not wolfram mathematica.
Please also tell me whether such numbers correspond to any famous known series.
Many are telling that there can be no such coefficients, then for part 1, please check $5x^2-5x+1=0$. It has it's roots between 0 and 1, and the coefficients are natural numbers to be sure, namely $a=5,b=5,c=1$.