When would not having a parabola x- intercept happen?

Answer 1

Consider a parabola described by the expression y = ax2 + bx + c

first, calculate it's first and second derivatives:

y' = 2ax + b

y'' = 2a

Find x value at which y' = 0, and calculate whether the corresponding y-coordinate is above or below the x-axis.

If it's above the x-axis, then the parabola will not intercept the x-axis if y'' is greater than 0.

If it's below the x-axis, then the parabola will not intercept the x-axis if y'' is less than 0.

Otherwise, it will always intercept the x axis at two locations.