Roots of a graph question (I have attached a picture of Q)

[Guest]

Please answer the question to the pic attached, thanks. I believe one of the possible values will be K = - 4 but not sure about the other one.

[HallsofIvy]

I hate having to turn sideways to read a problem. And it probably would have been easier to type the problem in rather than take a picture of the problem and upload it!
The problem is, simply, "The point with coordinates (-4, 0) lies on the graph of y= (x- k)^3- (x- k)^2+ 9(x- k). Find the two possible values of k."
Yes, k= -4 is one of the "possible values of k". But then we have a problem! Setting x= -4, y= 0, in this gives us a cubic equation for k. Clearly k is expected to be real and a cubic equation has either 1 or 3 real roots! And if we allow k to be non-real then there are 3 values. The cannot be "two possible values for k"!

[HallsofIvy]

Looking at this again, -4 is the only real value of k for which (-4, 0) is on that graph. There are NOT "two possible values of k".