Question 421852
For these types of problems, it often helps to graph the functions for some values of q (even though it is quite possible to solve without graphing).


{{{graph(400,400,-10,10,-30,30, 3-4x, 5x^2 - x, 5x^2 - x + 5, 5x^2 - x - 5)}}}


We can see from the graph that there is some constant q = C such that the graph of {{{y = 5x^2 - x + C}}} intersects {{{y = 3 - 4x}}} at exactly one point. Also, when q < C, the graphs will intersect at two points, and for q > C, the graphs do not intersect. We wish to find the value of C.


This occurs when {{{5x^2 - x + C = 3 - 4x}}} --> {{{5x^2 + 3x + (C-3) = 0}}}. The graphs intersect at only one point when the discriminant is zero, i.e.


{{{9 - 4(5)(C-3) = 0}}}

{{{9 - 20(C-3) = 0}}}

{{{9 - 20C + 60 = 0}}}

{{{20C = 69}}}, {{{C = 69/20}}}.


Therefore, all values {{{q > 69/20}}} satisfy.