Consider the quadratic equations y =3x^2 - 5x, y = 2x^2 - x - c where c is a real constant. (a) for what value(s) of c will the system have exactly one solution (a, b)? (b) for what value(s) of c will the system have more than one real solution? (c) for what value(s) of c will the system have no real solutions?

Consider the two functions as
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.

Do you need with the x value for the minimum? its always between the two intercepts for a parabola