# What are sources for non-routine problems involving quadratic functions (in one variable)?

I'm planning to get some sources which explain beautiful problems about quadratic function. I know that there are another kind of functions, but the quadratic function has different applications in some contexts such as maximum and minimum. Also this subject is perfect for high school student.

If the mathematical olympiads considered this topic as a part of the contest, that'd be great because in those events the problems are so amazing and help the students improve their skills about math.

I've read some books, but the most of them show us usual problems. However, there others betters. For instance, "Functions and Graphs", an english translations of Gelfand's book, which contains a lot of no-routine problems. Those problems could be a model to understand what I'm looking for.

• To clarify, are you talking about quadratic polynomials in one variable? If so, their theory is simple enough that essentially everything which could be known is already taught in 8th grade or so in the US (everything is made apparent by completing the square). Are you talking about more general quadratic forms? – Steven Gubkin Aug 22 '15 at 22:26
• I'm reading this as: What are sources for non-routine problems involving quadratic functions (in one variable)? If not a "great book," then maybe individual problems? – Benjamin Dickman Aug 22 '15 at 22:58
• Examples: 1. A quadratic curve $y = ax^2 + bx + c$ passes through two points $(−2,4)$ and $(4,4)$. Find the range of values of $a$ such that the curve has a minimum point above the $x$-axis. 2. Find the range of values of $a$ for which the quadratic expression $(a-2)x^2 - ax + (2a+3)$ is positive for all real values of $x$. – Benjamin Dickman Aug 22 '15 at 23:05
• What do you call a usual problem? If you gave an example it'd help provide ideas for you. – Karl Aug 23 '15 at 15:57
• @BenjaminDickman Yes, you're right. I've read "Functions and Graphs" an english translations of Gelfand's book. In that resource, there are beautiful no-routine problems and I'm looking for similar books. – Carlos Torres Aug 26 '15 at 23:44

I left a couple of problems (and their source) in an earlier comment.

Here are a couple more:

I'm not sure about the original sources for either of the following, but they should fit some interpretation of non-routine problems involving quadratic functions.

1. I gave an example in an answer to The philosophy of change of variables (MSE 226343) that we might as well update for next year:

$$(x-1)(x-2)(x-3)(x-4) = 2016$$

Observe $(x-1)(x-4) = x^2 - 5x + 4$ and $(x-2)(x-3) = x^2 - 5x + 6$. If you use a $u$ substitution for the former product, then the problem becomes $u(u+2) = 2016$, which is quadratic; so you can solve for $u$ and then solve for $x$ (though you might pick a nicer number than $2016$ to set this problem up!).

An alternative method of solution is to observe symmetry around $x = 3/2$; use a substitution to shift the graph $3/2$ to the left and you end up with a function symmetric about $x= 0$, i.e., even. A similar substitution reduces the problem back to a routine quadratic situation, and, in fact, the substitutions in both of the above approaches end up being equivalent.

1. Suppose you have $60m$ of fence and want to build a rectangular fence that maximizes the enclosed area. What are its dimensions? Okay, you have probably seen this problem enough times for it not to be considered non-routine. But how about a slight variation: Suppose you have access to a very long wall that can serve as one side of the rectangular fence. Again, build a rectangular enclosure that maximizes the enclosed area. Guess the dimensions, and then confirm whether your intuition was correct.

The first problem is, indeed, solved by a square: Each side is $15m$. You can tackle this by labeling adjacent sides as $x$ and $30-x$, so the area is $x(30-x)$; maximizing this downward facing parabola amounts to findings its vertex, which is why this is a quadratic problem.

An alternate solution, equivalent to the AM-GM inequality: The area of the $15m \times 15m$ rectangular enclosure is $225 m^2$; if we alter the dimensions, it makes use of additive inverses. That is to say, if we change one of the sides to $15+x$, then the adjacent side changes to $15-x$ in order to maintain a perimeter of $60$ (all measured in meters). But $(15+x)(15-x) = 225- x^2$ is clearly maximized when $x=0$ (i.e., in the aforementioned square case). In particular, a change to one of the sides by $x$ will decrease the enclosed area by $x^2$.

For the modified version, you now have access to the wall. So you can use all $60m$ to build the remaining three sides of your rectangular closure. Clearly, then, we end up with a square of dimensions $20m \times 20m$ with total area $400 m^2$. Right?

(Wrong. Why? Coming up with an "intuitive" explanation for what is the right answer perhaps qualifies as falling outside of the routine canon around quadratic function problems.)

• Students should be also able to find tangent lines to parabolas using only algebra. This involves finding, out of all of the lines passing through a given point on the parabola, which only has one point of intersection with the parabola. – Steven Gubkin Aug 31 '15 at 20:54
• Maybe this could be followed up with some play with reflective properties of parabolas. – Steven Gubkin Aug 31 '15 at 20:55
• @StevenGubkin I've been thinking about trying to scaffold an intuitive approach to derivatives by considering tangent lines to parabolas. Something like: (1) convince yourself tangent-line slopes correspond to a function; (2) convince yourself the function is only zero once, and (for lead coefficient positive) increasingly negative to the left of the zero, and increasingly positive to the right; (3) from (2) guess that the function could be a line; (4) note the zero is at the vertex, which occurs when $x = -b/2a$ so a non-crazy guess is $T(x) = 2ax + b$; check some examples by hand! Etc. – Benjamin Dickman Aug 31 '15 at 21:11
• (cont'd) Also, note that for the parabola $x \mapsto ax^2 + bx + c$ the constant added on, $c$, does not appear in our guess at $T(x)$; this is a good sign since the $+c$ has no effect on the shape of the parabola (only its position in the plane) hence should not appear in the tangent-line slope function $T(x)$... – Benjamin Dickman Aug 31 '15 at 21:14
• This is nice. Although, I think that they could actually calculate some slopes using the algebraic method at a few points, and then just do it for a general point. – Steven Gubkin Aug 31 '15 at 21:20