Algebra

Is There More Than One Standard Form for an Equation?

A couple recent questions asked what constitutes “standard form” for a quadratic equation; that will lead us to some older questions about “standard form” for a linear equation. We’ll see that “standard” isn’t quite as standard as you might think.

A Cubic Challenge

Let’s look at a nice little challenge: to find a cubic function with maximum and minimum at given locations – without using calculus. We’ll explore how to solve it with graphing software, and using algebra in a couple ways, and finally with calculus. And, surprise! They all give the same answer, though the results look …

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What If: Inventing “Pseudo-Complex” Numbers

We’ll work through a textbook exercise that encourages students to discover what it’s like to invent a new number system, as well as why some ideas work but others do not. The topic: What would happen if we changed the definition of the imaginary unit i so that its square is 1 rather than -1?

A System of Exponential Equations

What do you do when you are given a problem that starts with a “lie” and ends with a wrong answer? We’ll go in several directions with this problem, a system of two exponential equations in two variables.

Fun With a Functional Equation

Looking back at interesting questions I had to skip over when there were too many to choose, I found this interesting discussion of a functional equation.

Proving a Radical Expression Is Rational

It can be tricky deciding how to approach a proof; this problem, whose answer requires going in a very different direction than you might expect, provides some interesting insights into the nature of proof. The proof itself, in fact, is far less interesting than the process of getting there!

Finding a Locus: Algebra and Geometry

Last time we looked at the meaning of the concept of locus. This time, we’ll explore seven examples, from two students. We’ll look at both algebraic (equation) and geometric (description) perspectives.