Were you able to follow the argument? Probably the step from equation (1) to equation (2) presented no difficulties. But what about the next step? Do you see where equation (3) came from? If so, good for you. Most students, in fact most mathematicians, cannot make that step in their heads. But are you sure? Is there, perhaps, some small detail you’ve overlooked?
Check to see.
That is, let \(x=y-\frac{b}{2a}\) in equation (2) and see if you can get equation (3). Do it on that handy pad of paper we told you to get out earlier. Do it now. We’ll wait.
Done? Good.
Perhaps you haven’t been able to fill in the details on your own. That’s ok. Many people can’t. If not then get help: from a classmate, a friend, your instructor, whomever. Unfortunately most people won’t get help in this situation. Instead they will perceive this as “failure,” hide it and berate themselves or the problem as “stupid.” In short they will let their personal insecurities and demons overwhelm them. Don’t do this. Get help. You are neither dumb nor incapable. There are a thousand reasons that on any given day you might not be able to solve this problem. But don’t let a bad day interfere with the education you are here for. Get someone to help you over this hump. Later you will be able to help your helper in the same way. Really.
At this point we assume that you’ve successfully negotiated the transition from equation (2) to equation (5).
See? It really wasn’t that bad after all. Just a lot of elementary algebra. Now that you’ve done it (or seen it done), it is easy to see that there really wasn’t much there.
But this is the point! We left those computations out precisely because we knew that they were routine and that you could fill in the details. Moreover, filling in the details yourself gives you a little better insight into the computations. If we’d filled them in for you we would have robbed you of that insight. And we would have made this book longer than it needs to be. We don’t want to do either of those things. If we fill in all of the details of every computation for you, you won’t learn to have confidence in your ability to do them yourself. And this book will easily double in length.
So the lesson here is this: Keep that pad of paper handy whenever you are reading this (or any other) mathematics text. You will need it. Routine computations will often be skipped. But calling them “routine” and skipping them does not mean that they are unimportant. If they were truly unimportant we would leave them out entirely.
Moreover “routine” does not mean “obvious.” Every step we took in the development of the Quadratic Formula was “routine.” But even routine computations need to be understood and the best way to understand them is to do them. This is the way to learn mathematics; it is the only way that really works. Don’t deprive yourself of your mathematics education by skipping the most important parts.
As you saw when you filled in the details of our development of the Quadratic Formula the substitution \(x=y-\frac{b}{2a}\) was crucial because it turned
where \(k\) depends only on \(a\text{,}\)\(b\text{,}\) and \(c\text{.}\) In the sixteenth century a similar technique was used by Ludovico Ferrari (1522-1565) to reduce the general cubic equation
where \(p\text{,}\) and \(q\) depend only on \(a\text{,}\)\(b\text{,}\)\(c\text{,}\) and \(d\text{.}\)
The general depressed cubic had previously been solved by Tartaglia (the Stutterer, 1500-1557) so converting the general cubic into a depressed cubic provided a path for Ferrari to compute the “Cubic Formula” — like the Quadratic Formula but better.
Ferrari also knew how to compute the general solution of the “depressed quartic” so when he and his teacher Girolomo Cardano (1501-1576) figured out how to depress a general quartic they had a complete solution of the general quartic as well.
Alas, their methods broke down entirely when they tried to solve the general quintic equation. Unfortunately the rest of this story belongs in a course on Abstract Algebra, not Real Analysis. But the lesson in this story applies to all of mathematics: Every problem solved is a new theorem which then becomes a tool for later use. Depressing a cubic would have been utterly useless had not Tartaglia had a solution of the depressed cubic in hand. The technique they used, with slight modifications, then allowed for a solution of the general quartic as well.
Keep this in mind as you proceed through this course and your mathematical education. Every problem you solve is really a theorem, a potential tool that you can use later. We have chosen the problems in this text deliberately with this in mind. Don’t just solve the problems and move on. Just because you have solved a problem does not mean you should stop thinking about it. Keep thinking about the problems you’ve solved. Internalize them. Make the ideas your own so that when you need them later you will have them at hand to use.
Problem1.2.3.
Find \(M\) so that the substitution \(x=y-M\) depresses equation (6), the general cubic equation. Then find \(p\) and \(q\) in terms of \(a\text{,}\)\(b\text{,}\)\(c\text{,}\) and \(d\text{.}\)
Find \(K\) so that the substitution \(x=y-K\) depresses the general quartic equation. Make sure you demonstrate how you obtained that value or why it works (if you guessed it).
Find \(N\) so that the substitution \(x=y-N\) depresses a polynomial of degree \(n\text{.}\) Ditto on showing that this value works or showing how you obtained it.
Problem1.2.4.Another Derivation of the Quadratic Formula.
Here is yet another way to solve a quadratic equation. Read the development below with pencil and paper handy. Confirm all of the computations that are not completely transparent to you. Then use your notes to present the solution with all steps filled in.
Suppose that \(r_1\) and \(r_2\) are solutions of \(ax^2+bx+c=0\text{.}\) Without loss of generality suppose that \(a>0\text{.}\) Suppose further that \(r_1\ge r_2\text{.}\) Then
