That's why we invented word problems.
That's why we invented the Cartesian Coordinate System.
I know what the quadratic formula is, and how it is derived. What I would like to know is what it represents. For instance, if the solution to a quadratic equation is x=2 and x=6 how could I best conceptualize what these numbers mean? (As opposed to mechanically slogging through the equation without visualizing what the numbers represent)
I am so lame when it comes to explaining stuff
Thanks Bacterius. I am a complete math newbie so i appreciate your simple but effective explanation. Using the previous x=2 and x=6 might offer a binomial solution of (x-2)(x-6), correct? Why would putting in the x intercepts yield a value of 0 in each binomiaL. I haven't graphed a quadratic equation yet, so it's a bit difficult for me to visualize.
But what do you mean by each binomial? There is only one.
It's a model. A quadratic equation is not magic. It can be used as a model for many, many things. There are just too many to enumerate. As a model, it can represent anything! It is better for some things and worse for others. It's a pretty good model for throwing a ball and an okay model for building a remedial suspension bridge or an arched bridge. It's not a perfect model for anything unless we make it up. It's a pretty bad model for long term interest accumualtion of a bank account, but not terrible for particularly short term accumulation.
It's a tool. A quadratic equation is a tool. It can by used as a notational convenience to solve many, many problems. There are just too many to enumerate. I once built a printer stand from an existing structure in the shape of a rectangular prism. Deciding where to cut, fold, and reshape, I ended up with a quadratic equation. The notation itself made my life easier.
I certainly stand by my first comments. Find ANY algebra book and find ANY set of so-called "Word Problems". You will find all the practical applciations you possibly could want. Also, once you see the graph and get a little experience with it, you will begin to see related patterns all around you. The "Parabola" has interesting accoustical reflective properties. If you polish up a parabolic mirror, there are interesting optical properties.
My real point, though, is to forget about whatever might stop you from proceeding and learning. Take this as a promise. If you do EVERY problem in an Algebra book, you never again will ask what a quadratic equation represents. You will have seen so many applications and representations that it will become part of you.
I would also suggest that you lose the concept that "mechaniclly slogging" is not inherently valuable. If you are working on a problem and encounter a quadratic equation, don't you want to know how to handle it? You will not be able to predict every situation where you will encounter such a thing. A general idea how to deal with them will make you a cut above your neighbors. Practice more mechanical slogging, not just with quadratic equations, but with many other things, mathematical and non-mathematical alike.As opposed to mechanically slogging through the equation without visualizing what the numbers represent