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 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