yeah, you could isolate for a single instance of x after foiling the polynomial (vocab points!), then use the substitution method to plug that back in for every instance of x in the long form, making an ever messier equation, and solve again. i do think going to the quadratic is the best method.
the conic sections and their canonical forms of quadratics (shown very nicely by rod as y=ax^2+bx+c) are very powerful visual and mathematical concepts. you realize you can accurate plot all sorts of things fairly accurately with simple equations using parabolas, circles, and ellipses. hyperbolas are kinda their own thing almost, but cool for certain theoretical concepts. just taking a plane at different angles thru a three dimensional cone and measuring the x,y at certain points gives you all sorts of information for things, from the flight a bullet to the flight of the planets and stars.
that math is algebra, but some people might have learned in a "analytical geometry" and/or "advanced algebra" course. then you use similar math in calculus to measure the tangent, or slope at a particular point on the curve. given a few of these, you can approximate values and trends, approximate areas of complex surfaces with simple equations, find absolute mins/maxs, all sorts of very applicable math.
i think a lot of people who "don't like math" just had crappy teachers who didn't make the connection from "numbers on paper" to real world three dimensional concepts and applications. like a lot of teachers just kinda *said* hey you need the math because so and so uses it in real life for their job! but they didn't say *how*.
and then you can get into really fun stuff, like how those conic sections can be used to represent events in our lives! if you graph two cones, point to point and place their union at the origin of a 3d cartesian system, all the points with negative values for z will have occurred in the past. all points with positive z values will occur in the future. the origin is the event occurring right now in the x,y point in spacetime. the event taking place at the origin, right now, could not have occurred without all events in the negative(past) cone also having occurred. and also, each event in the positive(future) cone are events that possibly *could* occur in the future, given that event being measured at 0,0,0 has occurred. you could easily translate these to 2d coordinates.
so, that random polynomial that tanis made to demonstrate a point could potentially be used to measure, mathematically, how you tripping on the sidewalk led to you getting married. or something equally seemingly pointless in causality.
