Question

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

Answer 1

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