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.If I remember right there are 3 ways to solve an equation in quadratic form ax^2+bx+c. You can try to factor something out and make it simpler and then solve which can only be done sometimes, the aforementioned FOIL method and the equation method which always works just plugging in values. So after that I always found the other two a waste of time.
I remember thinking that was the end all hard stuff when I took algebra and cussing quadratic equations and then you get to calculus years and end up praying for them because the solution is always in the form of a nice simple Ax+b and didn't take half a page to get to something you could solve. You got so good at doing that stuff you did the algebra and trig part in your head believe it or not.
But man I have slept since them and that was 40 years ago. I guess that is a testament to those old battleaxe math professors I had that I can even recall anything at all. That was back in the day where we didn't get participation trophies and it was 3 tests and a comprehensive final ALWAYS. You had to get it and keep up or get left behind.
I personally always found statistics the most useful in my life for what I needed of the math I was exposed to. Even that stuff didn't make sense until one day an old Professor in a non math class who retired from Firestone took a bunch of data, made a dot plot that was trending upwards and to the left on a x, y axis. He slapped a line down the center turned the paper sideways and drew a bellcurve over the middle and it was like a light bulb finally went of in my head. Hey, this stuff is bad ass. Then I got real interested.
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.