I have a little brother who is in high school and he just learnt the quadratic formula for finding roots of second degree polynomials.

He asked me what why we learn this and how this could apply to a situation in the real world (preferably an application that would apply to his mindset).

I mentioned that the headlight of his bicycle is a parabola, but I was unable to explain why solving a second order polynomial equation like $x^2+3x-2=0$ would apply to this situation.

Could someone help me with some examples?

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It promotes thinking. –  Alizter Aug 27 '13 at 21:57
The main application, as such, I know is determining the behavior of second-order linear (homogeneous, constant-coefficient) ODEs, e.g. those describing circuits. –  Qiaochu Yuan Aug 27 '13 at 21:58
Hey, you didn't ask for pracical applications that can be understood using concepts he already knows about. That's really not how applications work in general. –  Qiaochu Yuan Aug 27 '13 at 22:06
I think the first application most are exposed to is physics. We know acceleration downward is constant (to first approximation) by Galileo, and add some velocity in some direction. The ball to this model follows a parabola, and where it hits the ground is the next zero of the equation. –  ex0du5 Aug 27 '13 at 22:10
Tell him he’s asking the wrong question. Tell him he should be asking how to do the same thing for polynomials of degree $3$. Point out to him that he didn’t ask that when somebody tells him to read a poem. Tell him that high-school mathematics was a weapon used for world domination (I’m thinking of trigonometry, especially the spherical kind, which was high-school stuff when I [voice creaks] was a boy). –  Lubin Aug 27 '13 at 22:40

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.

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!

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@AlexR. Hanging cables are catenary shapes. Hanging cables that are loaded uniformly with horizontal distance, with loads large compared to the cable, as in the bridge pictured by Maesumi above, are parabolas... –  DJohnM Aug 28 '13 at 2:22

Let $T$ be a triangle and denote the length of its hypotenuse by $x$, and the length of its other two sides by $x-a$ and $x-b$. Then what are the lengths of its sides as a function of $a$ and $b$?

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A quarterback throws a spiral to a wide receiver. The ball travels in the air along a parabola in such a way that after having flown $x$ (feet) horizontally it is at a height (in feet) $$y=\frac12x-\frac1{180}x^2$$ above the level of quarterback's shoulders. The receiver can beat the covering defender by jumping in the air so that his hands are $3$ feet above that reference height at the time he makes the catch. How far from the quarterback should he make his leap (assume that they get the timing right)? The equation has two solutions. Interpret both of them.

If I did the physics correctly, this corresponds to a ball thrown at $60$ ft/sec to an angle $\arctan(1/2)$ subject only to gravitational downward acceleration of $32$ feet per second per second.

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It is a mistake to believe that mathematics is about applications. It isn't. Mathematics is about the genesis and nature (properties) of its objects.

Once it was realized that every polynomial can be written as a product of linear binomials of the form (a_i x − b_i), the question arose of how to find the a_i and b_i, given the power-functions-series form which is a unique ("canonical") form of a polynomial (among others). Rewriting the polynomial as a constant times a monic polynomial can help a bit by making the a_i all ones and the b_i the roots. Converting the linear-binomials-product form to the power-functions-series form was was easy; one need only multiply and collect terms. But how to find the roots of the monic power-functions-series form was a question that intrigued people and was the source of challenges and contests. A number of special cases were discovered before someone finally derived the simple general formula for the roots of a quadratic (which is easy to derive once you see the idea), then the less simple cubic and quartic. Despite great efforts no one succeeded at finding a formula for the roots of a quintic or higher degree until eventually it was shown that for degrees greater than four, there can be no general formula, which is very surprising. Even so, at least some roots of certain quintics (and above) can be converted by formula, for example the real root of x^5 − 1, which enables us to divided the quintic by x−1 to obtain a quartic.

Polynomials have a number of curious properties.

All of mathematics is developed by raising and pursuing these kinds of questions. This was put very succinctly by Hannibal Lector in "The Silence of the Lambs" when he advised Clarise Starling to, "Read Marcus Arelius. Ask, 'what is the thing in itself? What is its nature?'"

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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.

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