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 doubt whether saying that "the solutions to a quadratic equation are the x-intercepts of the parabola modelled by the quadratic equation" actually is what Lucian wanted. I think he wants to have an example of how these quadratic equations could be using in practical applications. I'll give an example : how would you investigate the movement of a ball falling from some height, with a little initial force pushing it forwards ? Its movement (graph of height by time) sort of forms a parabola, and the quadratic equation and all the tools revolving around it allow easy and quick analysis of this parabola. In my example, one solution will be ignored (negative time), and another will give the time when the ball hits the floor.
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.
Perhaps I don't understand your question. "Why would putting in the x intercepts yield a value of 0 in each binomial"? Obviously, if you put x= 2 or x= 6 in (x- 2)(x- 6), one of the factors is will be 0 and 0 times anything is 0.
But what do you mean by each binomial? There is only one.
Okay, I'll give it another crack.
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