New math of the 70-ies. Yikes, or nostalgy ?

[symbolipoint]

The high school elementary and intermediate Algebra of the mid 1970's, at least where I was, did not really show much with solving cubic equations. Instead, those of us in Elementary Functions/PreCalculus dealt with cubics and higher order polynomials with Remainder and Rational Roots theorems, and the use of synthetic division, and Descartes Law of Signs. PreCalc also examined limits of functions. Much of what you showed in #35 about quadratic equations and functions (with your two roots, r and s) was also part of elementary & intermediate Algebra - both in high school and in college, but we were not shown that X=u+v substitution. That was clever.

About two or three years ago, I tried on my own to cube a binomial and look for corresponding coefficients, but I did not get the kind of results I hoped for. I did not know about that X=u+v substitution. I was making things more complicated too quickly in trying to use something like (x+k)^3. So far, having used three different College Algebra & PreCalculus textbooks during the last several years, I have not seen any kind of general solution to a cubic equation nor any derivation like you showed in post #35.

Interesting - last night I tried looking at the result for r+s, and (r)(s). Neat, simple expressions using a, b, and c from the general form ax^2+bx+c=0. Not certain if those resulting simple expressions have practical value, but interesting.

 

[symbolipoint]

If roots of quadratic equation are r and s, then r+s = -(b/a); rs = c/a,
For equation of the form ax^2 + bx + c = 0

(The forum's typeset formatting tool seems to be missing otherwise I hoped to typeset those.)

 

[mathwonk]

the brilliant insights i have shared here on quadratic and cubic equations are due to lagrange and euler. i recommend reading their algebra books! after more than 40 years of teaching undergraduate and graduate algebra i realized knew nothing compared to these giants.