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March 2003 Contents

**How (Not) to Solve Quadratic Equations**

Yves Nievergelt

Solving *ax*^{2}+ *bx* + *c* = 0 should present no difficulties, should it? Perhaps not, but it can. Such equations must be solved in the world beyond the classroom (which, by the way, is no more “real” than that of the classroom) and not all algorithms give the same result.

**Tangent Line Transformations**

Steven Butler

The geometric transformation of mapping a curve (*x*(*t*), *y*(*t*)) to the curve (slope of tangent line at *t*, *y*-intercept of tangent line at* t*) has a surprising property.

Recursive Enumeration of Pythagorean Triples/B>

Darryl McCullough and Elizabeth Wade

First comes (3, 4, 5), then (5, 12, 13) - then what? (7, 24, 25) or (8, 15, 17)? Here is a one-to-one mapping from pairs of positive integers to the set of all Pythagorean triples.

**Independent Thinking**

Reuben Hersh

2, 4, 8, 16; what comes next? In this triologue Ludwig answers 16, Imre answers 2. The question of how long the sequence continues is also considered.

**The Eccentricity of a Conic Section**

Ayoub B. Ayoub

What is the eccentricity of the conic section whose equation is *fx*^{2} + 2*gxy* + *hy*^{2} + 2*kx* + 2*ly *+* m* = 0? As you might expect, it isn’t anything extremely simple, but it isn’t all that complicated either.

**Lissajous Figures and Chebyshev Polynomials**

Julio Castiñeira Merino

Lissajous figures are those pretty curves generated by the parametric equations

*x* = *a* cos(*bt* + *c*), *y* =* d* cos(*et* + *f*).

It’s possible to eliminate the parameter and get (x, y) equations for them in terms of, of all things, Chebyshev polynomials

**Tossing a Fair Coin**

Leonard Lipkin

As long as probability endures we will be tossing coins, and as long as students endure some will fail to grasp what happens when we toss them for a long time. We know that the difference between the number of heads and the number of tails gets large, and it is good to remind students of this.

**Fallacies, Flaws, and Flimflam**

Ed Barbeau, editor

FFFs #204-#209, including a series that both converges and diverges.

**Classroom Capsules**

Warren Page, editor

**A Dozen Minima for a Parabola**

Leon M. Hall

Let the normal line to *y* = *x*^{2} at *P* intersect the parabola at *Q*. Here are twelve minimization problems based on this simple figure, with answers in numerical order.

**A Remark on the Chain Rule for Exponential Matrix Functions**

James H. Liu

Alas, the chain rule does not always hold when differentiating functions that involve e to a matrix power.

**A Codeword Proof of the Binomial Theorem**

Mark Ramras

A proof by counting. Clever counting, of course.

**Column Integration and Series Representation**

Thomas P. Dence and Joseph B. Dence

Tabular integration by parts gives some results beyond antiderivatives.

**Constrained Optimization with Implicit Differentiatio**

Gary W. De Young

A way of looking at maximum-minimum problems that has advantages over the usual method.

**Problems and Solutions**

Benjamin Klein, Irl Bivens, and L. R. King, editors

**Media Highlights**

Warren Page, editor

**Miscellanea**