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 this is the range of my root (Posted on 2016-10-14)
If ax^2 - bx + c = 0 has two distinct and real roots lying in the interval (0,1) where a, b and c are positive integers, then find the minimum value of abc.

 No Solution Yet Submitted by Danish Ahmed Khan No Rating

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 analysis then computation Comment 1 of 1
As this is an open interval, the inequalities below are strict.

x = (b +/- sqrt(b^2-4*a*c))/(2*a)

0 < b/(2*a) < 1   as the midpoint between the roots
must be between 0 and 1 inclusive
0 < b < 2*a

4*a*c < b^2 < 4*a^2  as the roots must be real, and a
repetition of previous inequality

c < a                implied by previous inequality

sqrt(b^2-4*a*c) < b  as the lower root must be positive

(b+sqrt(b^2-4*a*c)) < 2*a  as the upper root must be less than 1

a=5; b=5; c=1  works

With these coefficients, the roots are

x = (5 +/- sqrt(5)) / 10 ~= 1/2 +/-  0.223606797749979

BTW, if the interval had been closed, a=2; b=3; c=1 would have worked, giving a product of 6, with roots at 1/2 and 1, the larger root requiring a closed interval.

Back to the current problem:

minimum a*b*c = 25

from

`a b c    a*b*c5 5 1     256 5 1     306 6 1     366 7 2     847 6 1     427 7 1     497 8 2     1128 6 1     488 7 1     568 8 1     648 9 2     1448 10 3     2409 7 1     639 8 1     729 9 1     819 9 2     1629 10 2     1809 11 3     29710 7 1     7010 8 1     8010 9 1     9010 9 2     18010 10 1     10010 10 2     20010 11 2     22010 11 3     33010 12 3     36010 13 4     52011 7 1     7711 8 1     8811 9 1     9911 10 1     11011 10 2     22011 11 1     12111 11 2     24211 12 2     26411 12 3     39611 13 3     42911 14 4     61611 15 5     82512 7 1     8412 8 1     9612 9 1     10812 10 1     12012 10 2     24012 11 1     13212 11 2     26412 12 1     14412 12 2     28812 13 2     31212 13 3     46812 14 3     50412 14 4     67212 15 4     72012 16 5     96012 17 6     122413 8 1     10413 9 1     11713 10 1     13013 11 1     14313 11 2     28613 12 1     15613 12 2     31213 13 1     16913 13 2     33813 13 3     50713 14 2     36413 14 3     54613 15 3     58513 15 4     78013 16 4     83213 17 5     110513 18 6     140414 8 1     11214 9 1     12614 10 1     14014 11 1     15414 11 2     30814 12 1     16814 12 2     33614 13 1     18214 13 2     36414 13 3     54614 14 1     19614 14 2     39214 14 3     58814 15 2     42014 15 3     63014 15 4     84014 16 3     67214 16 4     89614 17 4     95214 17 5     119014 18 5     126014 19 6     159614 20 7     196015 8 1     12015 9 1     13515 10 1     15015 11 1     16515 11 2     33015 12 1     18015 12 2     36015 13 1     19515 13 2     39015 14 1     21015 14 2     42015 14 3     63015 15 1     22515 15 2     45015 15 3     67515 16 2     48015 16 3     72015 16 4     96015 17 3     76515 17 4     102015 18 4     108015 18 5     135015 19 5     142515 19 6     171015 20 6     180015 21 7     220515 22 8     2640`

For a = 1 To 15
For b = 1 To 2 * a - 1
For c = 1 To a - 1
DoEvents
If b * b > 4 * a * c Then
If 4 * a * a > b * b Then
If (b + Sqr(b * b - 4 * a * c)) < 2 * a Then
If Sqr(b * b - 4 * a * c) < b Then
Text1.Text = Text1.Text & a & Str(b) & Str(c) & "     "
Text1.Text = Text1.Text & a * b * c & crlf
End If
End If
End If
End If
Next
Next
Next

Edited on October 14, 2016, 3:43 pm
 Posted by Charlie on 2016-10-14 15:41:10

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