Suppose you have the following quadratic equation to solve:like “x + 4” is a polynomial, but
polynomials can have one or
many variables in any combi-
nation, with their magnitude
determined by what power the
variables are taken to, like x^2.
That means a quadratic equa-
tion may be written out as
something like x^2 – 8x + 12 = 0.
Dr. Loh’s method found a sim-
pler way to derive the roots of a
quadratic equation, namely via
a shortcut through the tedious
guess-and-check method used in
factoring. Students traditionally
try to guess for two numbers that
have a certain sum and a certain
product, by guessing what the
factorizations of the product may
be. Loh realized he could instead
focus on the desired sum, look-
ing for two numbers equidistant
from the desired average (or half
the desired sum).
“The individual steps of this
method had been separately
discovered by ancient mathema-
ticians,” Loh said. That includes
the Babylonians and Greeks.
“The combination of these steps
is something that anyone could
have come up with.”
Prior to Loh’s method, the
closest anyone had come to
explaining this strategy was a
math teacher, John Savage, who
published an article in 1989 in
the journal The Mathematics
Te a c h e r that described a simi-
lar idea. Still, Savage’s idea used
different logic and an extra,
unnecessary step.
While it’s certainly deviant
from conventional math strat-
egy, Loh believes his method
brings a level of intuition that
will appeal to students, so long
as they remember some sim-
ple generalizations about roots
and how they relate within a
quadratic equation. It’s still
complicated, but it’s less con-
voluted, especially if Loh is
correct in thinking this will
help students tackle quadratic
equations that they’re more
likely to encounter in the real
world. That’s important. Mas-
tering quadratics is a precursor
to many precalculus courses.
Because outside of class-
room-ready examples, thequadratic method isn’t simple.
Real examples and applications
are messy, with ugly roots con-
sisting of decimals or irrational
numbers. As a student, it can be
hard to know you’ve found the
solution even in clean textbook
problems, but Dr. Loh’s new
method might bring simplicity
to both the learning environ-
ment and work environment at
the same time.Dr. Loh’s Method
“Normally, when we do
a factoring problem, we
are trying to find two
numbers that multiply
to 12 and add to 8,” Dr.
Loh said.
Those two numbers
are the solution to the
quadratic, but it takes
students a lot of time
to solve for them, as
they’re often using
a guess-and-check
approach.
Instead of starting by
factoring the product,12, Loh starts with
the sum, 8.
If the two numbers
we’re looking for,
added together, equal
8, then they must
be equidistant from
their average. So
the numbers can be
represented
as 4–u and 4+u.
When you multiply,
the middle terms
cancel out and you
come up with the
equation 16–u^2 = 12.When solving for u,
you’ll see that positive
and negative 2 each
work, and when you
substitute those integers
back into the equations
4–u and 4+u, you get
two solutions, 2 and 6,
which solve the original
polynomial equation.
It’s quicker than the
classic foiling method
used in the quadratic
formula—and there’s
no guessing required.
—Courtney LinderNeed product to = 12 // Need sum to = 8x
2
-8x+12=( x-2 )( x-6 ) =4 - u; 4 + u 16 - u^2 = 12 u^2 = 4 u = 2Solution = ( 2,6 )March/April 2020 11