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1) A quadratic will have no real roots when its discriminant is negative
Dividing by 4,
We have a positive coefficient on k^2 so this parabola is a CUP (concave up positive) so has a minimum at the vertex. If the vertex y value is less than zero, the inequality will be true in the range between the zeros.
That's true for
2) We look for the meet of the line and the parabola:
For two intersections we need a positive discriminant:
That's means two negative or two positive, so
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