I think that the question is incomplete.

It should be mentioned in the question that "the given equation must be zero for atlest one real value of x" or either that "the equation has 1/2/3 roots.

Then we can proceed to solve the problem by drawing graph of

x^3 - 9x^2 + 24x = 0.(Let it be f(x))

We can draw the graph easily by analysing it derivative and second derivative.

Then we can verify for what values of K in the codomain of f(x), there is one value of x or two or three values of x.