for x-intercept, y = 0. so we are concerned with the condition:

9x^2 + 3kx + k = 0

by the quadratic formula:

x = [-3k +/- sqrt(9K^2 - 4(9)k)]/2(9)

=> x = [-3k +/- sqrt(9K^2 - 36k)]/18

a) we have one x-intercept if 9K^2 - 36k = 0

=> 9k(k - 4) = 0

=> k = 0 or k = 4 for 1 x-intercept

b) for 2 x-intercepts, 9K^2 - 36k > 0

=> 9k(k - 4) > 0

=> k < 0 or k > 4

so to satisfy this condition, k has to be on the interval (-infinity, 0) U (4, infinity)

we had to test these values on a number line to know which way the inequalities should go

c) for no x-intercept, 9K^2 - 36k < 0

=> 9k(k - 4) < 0

=> 0 < k < 4

Do you know why these conditions have to be fulfilled the way they are?