Questions about solving algebraic equations are common. Here I will bring together several answers where we discussed the basic principles for solving relatively simple equations, which are important to learn well before moving on to quadratic equations and beyond.
I have often noted that calculus class is where you really learn algebra. Certain techniques in calculus demand algebraic skills that either were not taught in algebra classes (because they are not needed until you get to calculus), or have been forgotten. Chief among these is the method of partial fractions. I have here put …
(New Question of the Week) We recently had a long discussion about a very common question from a somewhat different perspective: What do exponents (zero, negative, fractional, …) actually mean? The hard part, in the end, was to decide what “mean” means. What does it mean to define something in math? I will pick out the main thread of the …
(New question of the week) A conversation last week went through a number of interesting questions, starting with a couple on percentages, and moving into some that I would call rate questions. I will extract these, which I think will be useful for others. (The rest could, too, but there was just too much there …
(Archive Question of the Week) Students commonly expect that textbooks all say the same thing (in fact, some think they can ask us about “Theorem 6.2” and we’ll know what they’re talking about!). The reality is that they can even give conflicting definitions, depending on the perspective from which they approach a topic. Here, I …
Definitions are interesting in several ways. Sometimes they are essentially arbitrary; other times there is a very good reason for them, and understanding that reason can be helpful in understanding and using them. But they are usually taught just as something to memorize. Let’s think about why slope is defined as it is, and not …
(New Question of the Week) Sometimes a problem leads to a very interesting discussion that brings out many good ideas – but then turns out to be something entirely different, which brings out even more (and simpler) ideas. This polynomial equation problem we helped with last week was like that. I will not be quoting …
(New Question of the Week) From time to time we get a question that is more about words than about math; usually these are about the meaning or origin of mathematical terms. Fortunately, some of us love words as much as we love math. But the question I want to look at here, which came …
Recall that the domain of a function is the set of all valid input values (x), and the range is the set of all possible output values (y). It is reasonably easy to find the domain: look for what could make it impossible to evaluate, such as dividing by zero or taking the square root …
(New Question of the Week) I like working with a student who is willing to take chances, and also willing to be corrected. As I have often explained, just like a medical doctor, a Math Doctor wants you to “show me where it hurts” in order to diagnose the problem; so showing detailed work is …