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If (ax + b)(bx + a) = 60x^2 + v * x + 60, where a,b , and v are distinct integers, what is the minimum possible value of v, the coefficient of x?

 Dec 3, 2020
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(ax + b)(bx + a) = 60x^2 + v * x + 60

 

(ab)x^2  +  ( a^2 + b^2)x  + (ab)  =  60x^2  +  vx  +  60

 

It's  obvious that   ab    =  60

And (a^2 + b^2)  =  v

 

Divisors  of 60   =

1 | 2 | 3 | 4 | 5 | 6 | 10 | 12 | 15 | 20 | 30 | 60 

 

The minimum  value  for  a^2 + b^2  =  v  will be produced  when  a = 6  and  b  =10

 

So

 

v = (6)^2 + (10)^2  =   136

 

 

cool cool cool

 Dec 3, 2020

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