Here are two views about how to symbolically represent two quadratic functions plotted in the {x,y} plane.
1) A pair of simultaneous linear equations of the form and , or
2) A single equation of the form .
If you regard the two linear functions as a pair of simultaneous quadratic equations then the solution set are the points in the {x,y} plane where the graphs of and intersect.
If you regard the two quadratic functions as a single equation of the form then the solution set are the points on the x axis whose x coordinates are the x coordinates of the intersection points of the two graphs.
Which formulation do you prefer ? Why ?
What problems could/would you pose to your students based on this applet ?