if a & b are the roots of a quadratic equation 2x^2 + 6x + k = 0, where k<0, then what is the maximum value of (a/b)+(b/a) ?

Let r be the lesser root and r^2 be the greater.......the sum of the roots   = -b/a  = -[-6] / 1  = 6
 
So we have that
 
r^2 + r  = 6     →    r^2 + r - 6 = 0      
 
Factor
 
(r + 3) (r - 2)   = 0  
 
So r = -3    or   r = 2
 
Then r^2  = 9     or r^2  =4
 
And the product of the roots =  c/a   =  k/a  = k
 
So....k =   (-3)(9)   = -27     or k = (2)(4)  = 8
 
Check
 
x^2 - 6x - 27  = 0      factors as  (x + 3)(x - 9) = 0     and  the roots are -3 and 9
 
x^2 - 6x + 8  = 0      factors as (x -2) (x - 4) =0      and the roots are 2 and 4