.
Determine the set of values of the constant k for which the line y = 4x + k does not
intersect the curve y = x^2
~~~~~~~~~~~~~~~~~
Here an Algebra solution is placed.
If the/an intersection point does exist, then an equation
x^2 = 4x + k (1)
has a real solution. This equation is equivalent to
x^2 - 4x - k = 0.
The discriminant of this quadratic equation is
d = b^2 - 4ac = (-4)^2 - 4*1*(-k) = 16 + 4k.
The discriminant is positive if and only if
16 + 4k >= 0
4k >= -16
k >= -16/4 = -4.
Thus a real solution to equation (1) does exists if and only if k >= -4.
If k < -4, there is no real solution for equation (1), so the intersection does not exist.
ANSWER. The set of values of "k" is { k < -4 }.
Solved.