In the equation x^2+kx+54=0 one root is twice the other root find k

Let r be the lesser root and r^2 be the greaterthe 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

 

Sok =   (-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