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If the quadratic equation a_(1)x^(2)-a-(2)x+a_(3)=0 where a_(1),a_(2),a_(3) in N has two distinct real roots belonging to the interval (1,2) then least value of a_(1) is_______

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Solution :`alpha+beta=(a_(2))/(a_(1)), alpha beta=(a_(3))/(a_(1)), alpha, beta epsilon (1,2)`
`implies alpha-1, beta-1, 2-alpha, 2-beta epsilon (0,1)`
A.M. `ge` G.M. `implies((alpha-1)+(2-alpha))/2 ge sqrt((alpha-1)(2-alpha))`
`implies(alpha-1)(2-alpha) le 1/4` SIMILARLY `(beta-1)(2-beta) le 1/4`
`0 lt (alpha-1)(2-alpha)(beta-1)(2-beta) lt 1/16` (Both can't equal to `1/4` simultaneously `(alpha !=beta)`
`0 lt ((a_(3))/(a_(1))-(a_(2))/(a_(1))+1)(4-(2a_(2))/(a_(1))+(a_(3))/(a_(1))) lt 1/16`
`0 lt ((a_(1)-a_(2)+a_(3))(4a_(1)-2a_(2)+a_(3)))/(a_(1)^(2)) lt 1/16`
`(a_(1)^(2))/16 gt (a_(1)-a_(2)+a_(3)(4a_(1)-2a_(2)+a_(3))`
`(a_(1)^(2))/16 gt f(1).f(2)implies(a^(2))/16 gt 1`
`implies` Least integral alue of `a` is 5


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