The archway of the main entrance of a college is modeled by the quadratic equation y=-x^2+6x the university is hanging a banner at an angle defined by the equation 4y=21-x. at what two points should the banner be attached

The two points represents the intersection of the line with archway 
to find these points, we solve it as follow
y=21/4-x/4 and  y=-x^2+6x
y=y implies 21/4-x/4 = -x^2+6x, so  21/4= -x^2 +6x+x/4 equivalent to 
21= -4x^2 +25x
we have to solve  -4x^2 +25x-21=0, this equation has exactly to solution x=1, and x=21/4
so the two points are  A(1,5) and B(21/4, 63/16)