If (ax+b)(bx+a)=26x^2+cx+26, where a, b, and c are distinct integers, what is the minimum possible value of c, the coefficient of x?
(ax+b)(bx+a)=abx2+(a2+b2)x+ab=26x2+cx+26ab=26c=a2+b2
so basically we're minimizing a2+b2 subject to ab=26and subject to a,b,c∈Z
26 can be factored as (2,13), (1,26)22+132=17312+262=676and 173 is the smaller of those two so c=173