Quadratic inequality with parameters

[mathwannabe]

Hello everybody

1) I got this problem:

"How many whole number values for the parameter k are there, so that (k - 1)x^2 - 2(k + 5)x - (k + 5) < 0 is true for every real x ?"

This is how did it. So, for this parabola to never intercept the x axis and to be negative on the y axis in its entirety, it has to be that D < 0 and a < 0.

I set up my D to be: (-2)^2(k + 5)^2 + 4(k - 1)(k + 5) < 0

Which led to: k^2 + 7k + 10 < 0

Which led to: -5 < k < -2

Which also meets the condition that a < 0 ... (k - 1) < 0

Which led me to my answer that there are TWO values for the parameter k {-4, -3} for which (k - 1)x^2 - 2(k + 5)x - (k + 5) < 0 is true for every real x.

That is also the answer that was provided in my source for this problem. As there is no workflow in my source, I have no idea if what I did was right, so I am asking you to check it out and tell me how badly I missed the target XD

Thank you in advance...

 

[mathwannabe]

It looks to me like a very sensible mode of procedure.

I just realized something that has gotten me really confused:

"How many whole number values for the parameter k are there, so that (k - 1)x^2 - 2(k + 5)x - (k + 5) < 0 is true for every real x ?"

But if I set my D to be less than zero, roots for the equation will be complex conjugates, isn't that right? The text of the problem states that "... so that (k - 1)x^2 - 2(k + 5)x - (k + 5) < 0 is true for every real x ?"

I am so confused right now...

... or I am just too tired to think

 

[mathwannabe]

Go to bed. You are tired.

You are being asked a question about real numbers so complex conjugates are irrelevant.

Your initial thought was fine. If the leading coefficient is negative, then the quadratic is negative for large real values of x. If the determinant of the quadratic is negative, there are no real roots so the quadratic never crosses the x-axis. Put those two constraints together, and the quadratic is everywhere negative. The only remaining task is the mechanical one of finding whether there are any whole numbers that satisfy both constraints.

I am sorry that my first answer was too curt.

Thank you Jeff Yes, I understand completely now!