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) ?

with solution plz

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  • 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
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