The use for solving quadradic equations for high school students

The quadratic polynomials have numerous applications in the physical world. Some of them have been pointed out above. There are many others: the intensity of a sound or light decreases with the square of the distance; gravitational attraction between two objects is inversely proportional to the square of the distance between them; orbits of planets are ellipses which are expressed as quadratics; and on and on. So learning about quadratic equations opens up the solution to many practical problems.

The quadratics also are the starting point for building up the theory of higher order polynomials, as well as lots of other mathematics in which squares appear. The higher order polynomials, which are essential for doing any higher mathematics, have many applications in the real world.

For example, if you have any interest in money (and unless you have a large trust fund you have to) you will find that higher order polynomials are very involved in the calculation of compound interest, payment streams and many other financial matters.

"Pure" as opposed to "applied" mathematics supposedly does not connect to real world problems. The idea is that one can explore aspects of mathematics which do not seem to have any practical applications. And some of those aspects are very complex, well-structured, and very interesting, just as any natural object -- a tree or a starfish -- can be very interesting.

It is also not true that "pure" mathematics is pure. An enormous amount of it turns out to have important applications, which are simply not obvious at the time; or which are in areas that were not yet developed. One might almost call pure mathematics pre-applied mathematics. It is there, and when it is needed it will be put to use.

With all that said, I think the questions about "relevance" are in some sense an excuse for ducking out of the harder work of learning. "It's not relevant so I won't bother with it". I would say EVERYTHING is relevant. We are entire people living in a complex world, and everything connects to everything else. We need to know as much as we can just to manage our lives.

Tell him mathematics prepares you "to go where no one has gone before" (that is a quote from Star TreK if you noticed!).

You can go to higher dimensions and you can come face to face with infinity. What could be more exciting?!

At his age kids want some reassurance and their question about "application" is something they learn to ask from grown ups. That is their way of expressing their nervousness. Or just to start a conversation, more like saying ``it is going to be a rainy day''.

Perhaps he needs a counselor. Perhaps he needs a tutor. Perhaps he needs a teacher who can put some pizzazz into things. Perhaps he better do his homework with a classmate, etc.

Many of my calculus students also say similar things (what is the application? they ask) but when I tell them you can explain a rainbow with calculus they do not show any interest in producing a project based on it.

The question would be considered honest if the student asks it across the board while showing good faith effort at doing the homework. Does he ask what is the use of a video game? What does ``useful" mean to him?

Perhaps if you monitor him while taking lessons from reputable online resources he could warm up to it. Khan Academy is popular these days. On the other hand tutoring or monitoring by a family member can easily lead to a power struggle, so you need to approach with caution and infinite patience. Best you can hope for is a positive peer influence or a good teacher.

Here is an uplifting portrayal of mathematics Geometry of Nature.

To be specific you can tell him that in order to find where a ball is going to land, after you kick it or throw it, you need to solve a similar equation. Here is more explanation.

Other applications of parabolas (the shape of $y=ax^2+bx+c$) or paraboloids (what you get when you rotate a parabola around its axis):

1- Parabolic mirrors and antennas (the dish you have on your roof is a paraboloid), Image showing the difference between a paraboloid and an ordinary curved or spherical reflecting surface.

2- The shape of mirrors used in search lights (those dancing lights you see in back of some trucks during grand opening day of a new store etc.) Image

3- A pale of water (or coffee cup) that is stirred takes the form of a parabloid (that is used for making mirrors that are then used in astronomy), Demo showing Newton's rotating pale of water experiment. (this needs a download)

4- The shape of hanging cable of traditional bridges. Picture. (Clarification: The main cable that holds the road, under certain ideal conditions, has the shape of a parabola. A free cable, as in a power cable hanging between poles in streets, has a different shape called a catenary. It resembles a parabola but it is very different. Also see the comment below.)

5- Trajectory of comets, Video. (Assuming the comet has just enough energy to approach and escape sun once.)

6- Trajectory of items thrown into air near Earth. (Assuming air friction and wind are not influential.) To see it clearly just hold a hose and let water run into air with some pressure. The trajectory of water is a parabola. To find out where the water stream strikes ground is same a solving a quadratic equation.

7- Parabolic Loudspeakers.

8- Acoustic Mirrors. Or search for "whisper dish".

Finally, there are many dishes on the buffet table of skills next to mathematics. Be happy if he has interest in some items. Many (most?) people live perfectly happy or successful lives without mathematics!