Natural-born Roots
I hereby bequeath to the world a formula that I have taught for thirty-five years
but have never seen in any textbook. I published it in The Oklahoma University
Mathematics Letter in 1963.
It relegates the traditional quadratic formula to secondary status because this
natural formula, which I call “McGuire’s Formula” relates inherently to the parabolic
graph of the quadratic and because it is less than half the size of the traditional formula.
The parabola existed in nature as the trajectory of an object slung into the
atmosphere long before any geometric or algebraic constructs were made by humans.
Hence, a formula that relates the roots to the vertex of the parabola is more natural than
the traditional formula, which expresses the roots in terms of the coefficients of a
deviously derived general form.
As figure 1 shows, where (h, k) is the vertex, the roots are simply
h  (-k/a).
This expression derives from the standard form of the quadratic, y = a(x – h)2 + k, letting
y be zero. It is equivalent to the traditional quadratic formula.
Figure 2 displays a survey of cases in which the roots of the quadratic are
desired. In three of the four cases, the natural formula is preferable, since, if a graph is
required, the vertex is one of the critical points and once that is obtained, the roots
come from there.
Article published in The Mathematics Teacher magazine, Vol. 90. No. 6, September
1997.
By H. Paul McGuire
phyl.hil@gmail.com
Miramar Community College
San Diego, CA 92126