Using the sum-product method:
Let a+b=x+7
and let ab=7x+3
then ab=7(a+b)-46
When (a+b-7)^2-(a+b)(a+b-7)+7(a+b)-46-3=0
Note that this is true for all {a,b} in that everything cancels.
So it is unnecessary to manipulate the given equation x^2-17x+73=0: all that matters is the coefficients: a+b=x+7, and ab=7x+3.
Some simple manipulation:
(ab+46)/7=a+b
(a(7a-46)/(a-7)+46)/7=a+b
b = 3/(a - 7) + 7
(b-7)(a-7)=3
giving the solutions: {a,b,x} = {4,6,3}{6,4,3}{8,10,11}{10,8,11} over the integers.
Of these, only the second pair meets the required 'base' criterion of the puzzle.
Edited on February 7, 2023, 10:56 pm
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Posted by broll
on 2023-02-07 09:44:24 |
Interesting
