I'm coming up with a strange result . . .
A curve has the equation: . where is a constant.
By considering . . . . I don't see how
. . find the range of such that the equation: .
. . has exactly 2 real roots for all real values of
The equation is: .
. . which simplifies to: .
A quadratic has two real roots if its discriminant is positive.
The discriminant is: .
. . And we want: . .
This is an up-opening parabola.
. . It will be positive if its vertex is above the -axis.
Its vertex is found at: .
. . Hence, the vertex is: .
The parabola is positive if: .
We have a constaint on
Yet we must find values of which satisfy  for all values of
Is there a flaw in my reasoning?