APPLICATION OF AVERAGING CONCEPTS: FACTORING A QUADRATIC
 

Basic concepts—when deeply understood—can often be applied in seemingly-unrelated and powerful ways.
In this lesson, simple averaging ideas lead to a method of factoring quadratics that is quick, easy, and reliable.
No quadratic formula! No guess-and-check! No time spent trying to find ‘numbers that work’!

This section is optional, and is for readers who have already studied factoring.
This section is not included in my Compilation of Web Page Lessons book.
There are no exercises in this section.

Po-Shen Loh discovered this technique.
Many thanks to Owen Mortensen for bringing it to my attention.

Learn by example? Jump right to them!

REVIEW OF AVERAGING CONCEPTS

Let $\,a\,$ and $\,b\,$ be real numbers.

REVIEW OF FACTORING CONCEPTS

FACTOR: $\,x^2 + bx + c\,$
Note:

THE NEW METHOD: ‘FOCUS FIRST ON THE SUM’

In its full generality, the ‘Focus First On The Sum’ method looks complicated—but it truly isn't!
A few examples show just how quick, easy, and versatile it is:

EXAMPLES:
Factoring Quadratics using the ‘Focus First on the Sum’ Method

Factors involve only integers



Factoring when the coefficient of the square term isn't $\,1\,$



Factors involve irrational numbers



Factors involve the imaginary number $\,i\,$

When you're done with this section, move on to:
Average of Signed Numbers