A Brief Review of Certain Topics in Algebra
I will largely illustrate various rules and techniques of algebra through the use of examples. I will not justify or prove all that I am doing. I'll leave that for the math teachers.
Does it bother you that in the problem above, using the quadratic formula, we solved for t and not x?
Maybe it did, and maybe it didn't. It does tend to make some students uncomfortable, at least at first.
The quadratic formula is often presented as:
This corresponds to the standard form of the quadratic equation ax2 + bx + c = 0, where a, b and c are constants, for which we hope to find the value of x.
Realize that the variables used here are arbitrary. There is nothing wrong with presenting the quadratic formula as:
nt2 + mt + p = 0 where n,m and p are constants, and
The quadratic formula, like all of the various formulas in algebra, speaks to relationships between variables and constants. The formula above preserves the relationship between the variable and constants, just as the version involving a,b,c and x does.
In the same vein, the equation of a line is often presented as:
where m and b are constants.
But there is nothing to prevent us from writing:
where v and c are constants. Here, we find a linear relationship between x and t, just as we found one between y and x before. Indeed, this is standard practice in physics, where specific variables are used to represent specific real-world quantities, e.g. "t" for time.