150 CHAPTER 5 ALGEBRA
The tactic of extracting squares includes many tools in addition to completing
the square. Here are a few important ideas.5.2.13 (x-y)^2 +4.xy = (x+y)^2.
5.2.14 Replacing the variables in the equation above by squares yields(�-i)^2 +4�i = (x^2 +i)^2 ,
which produces infinitely many Pythagorean triples; i.e., integers (a,b ,c) that satisfy
a^2 + b^2 = c^2. (In a certain sense, this method generates all Pythagorean triples. See
Example 7.4.3 on page 242 .)
5.2.15 The following equation shows that if each of two integers can be written as the
sum of two perfect squares, then so can their product:(x^2 +i)(a^2 +b^2 ) = (xa-by)^2 + (ya+bx)^2.
For example, 29 = 22 + 52 and 13 = 22 + 32 , and, indeed,
- 13 = 112 + (^162).
It is easy enough to see how this works, but why is another matter. For now, hindsight
will work: remember that many useful squares lurk about and come to light when
you manipulate "cross-terms" appropriately (making them cancel out or making them
survive as you see fit). For a "natural" explanation of this example, see Example 4.2. 16
on page 13 0.
Substitutions and Simplifications
The word "fractions" strikes fear into the hearts of many otherwise fine mathematics
students. This is because most people, including those few who go on to enjoy and
excel at math, are subjected to fraction torture in grade school, where they are required
to complete long and tedious computations such as"S. l"f"I 10 x^2 II
Imp I y
X _ I+
17 - x- 1 - 5x
+
3·
You have been taught that "simplification" is to combine things in "like terms."
This sometimes simplifies an expression, but the good problem solver has a more
focused, task-oriented approach, motivated by the wishful thinking strategy.
Avoid mindless combinations unless this makes your expressions sim
pler. Always move in the direction of greater simplicity and/or symme
try and/or beauty (the three are often synonymous).(There are, of course, exceptions. Sometimes you may want to make an expression
uglier because it then yields more information. Example 5.5.10 on page 175 is a good
illustration of this.)
An excellent example of a helpful substitution (inspired by symmetry) was Exam
ple 3.1.1 0 on page 69, in which the substitution y = x + 1/ x reduced the 4th-degree