Consider the two functions as <span>y1(x) =3x^2 - 5x, y2(x) = 2x^2 - x - c
The higher the value of c, father apart the two equations will be. They will touch when the difference, i.e. y1(x)-y2(x)=x^2-4*x+c has a discriminant of 0. This happens when D=((-4)^2-4c)=0, or when c=4. (a) So when c=4, the two equations will barely touch, giving a single solution, or coincident roots. (b) when c is greater than 4, the two curves are farther apart, thus there will be no (real) solution. (c) when c<4, then the two curves will cross at more than one location, giving two distinct solutions.
It will be more obvious if you plot the two curves in a graphics calculator using c=3,4, and 5.
When we add a constant to the end of a parabola, it shifts the graph up and down. Since the constant here is positive, the graph moves up by that number (5)
