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this is the range of my root (Posted on 2016-10-14) Difficulty: 4 of 5
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    
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Solution 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*c
5 5 1     25
6 5 1     30
6 6 1     36
6 7 2     84
7 6 1     42
7 7 1     49
7 8 2     112
8 6 1     48
8 7 1     56
8 8 1     64
8 9 2     144
8 10 3     240
9 7 1     63
9 8 1     72
9 9 1     81
9 9 2     162
9 10 2     180
9 11 3     297
10 7 1     70
10 8 1     80
10 9 1     90
10 9 2     180
10 10 1     100
10 10 2     200
10 11 2     220
10 11 3     330
10 12 3     360
10 13 4     520
11 7 1     77
11 8 1     88
11 9 1     99
11 10 1     110
11 10 2     220
11 11 1     121
11 11 2     242
11 12 2     264
11 12 3     396
11 13 3     429
11 14 4     616
11 15 5     825
12 7 1     84
12 8 1     96
12 9 1     108
12 10 1     120
12 10 2     240
12 11 1     132
12 11 2     264
12 12 1     144
12 12 2     288
12 13 2     312
12 13 3     468
12 14 3     504
12 14 4     672
12 15 4     720
12 16 5     960
12 17 6     1224
13 8 1     104
13 9 1     117
13 10 1     130
13 11 1     143
13 11 2     286
13 12 1     156
13 12 2     312
13 13 1     169
13 13 2     338
13 13 3     507
13 14 2     364
13 14 3     546
13 15 3     585
13 15 4     780
13 16 4     832
13 17 5     1105
13 18 6     1404
14 8 1     112
14 9 1     126
14 10 1     140
14 11 1     154
14 11 2     308
14 12 1     168
14 12 2     336
14 13 1     182
14 13 2     364
14 13 3     546
14 14 1     196
14 14 2     392
14 14 3     588
14 15 2     420
14 15 3     630
14 15 4     840
14 16 3     672
14 16 4     896
14 17 4     952
14 17 5     1190
14 18 5     1260
14 19 6     1596
14 20 7     1960
15 8 1     120
15 9 1     135
15 10 1     150
15 11 1     165
15 11 2     330
15 12 1     180
15 12 2     360
15 13 1     195
15 13 2     390
15 14 1     210
15 14 2     420
15 14 3     630
15 15 1     225
15 15 2     450
15 15 3     675
15 16 2     480
15 16 3     720
15 16 4     960
15 17 3     765
15 17 4     1020
15 18 4     1080
15 18 5     1350
15 19 5     1425
15 19 6     1710
15 20 6     1800
15 21 7     2205
15 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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