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.
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
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Posted by Charlie
on 2016-10-14 15:41:10 |
analysis then computation