Hello, kingman!

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: . .[1]

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 [1] forvalues ofall

Is there a flaw in my reasoning?