1. ## Quadratic Functions and Equations

I was assigned an achievement check a few days back and it is due soon.

Question: Find k such that the graph of y=9x^2 + 3kx + k:
a) intersects the x-axis in one point only
b) intersects the x-axis in two points
c) does not intersect the x-asix

I am unable to figure out how to solve it. Can someone explain to me how to do this please?

2. Originally Posted by DFKnight
I was assigned an achievement check a few days back and it is due soon.

Question: Find k such that the graph of y=9x^2 + 3kx + k:
a) intersects the x-axis in one point only
b) intersects the x-axis in two points
c) does not intersect the x-asix

I am unable to figure out how to solve it. Can someone explain to me how to do this please?
for x-intercept, y = 0. so we are concerned with the condition:
9x^2 + 3kx + k = 0

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?

3. Thanks a lot for the detailed explaination. And yes, I do understand why these conditions have to be fulfilled the way they are.