5.4 The Conjugate Rule
In chapter 8.7 in Operators and Calculation Rules, we covered the conjugate rule.

The conjugate rule is a rule in algebra that tells you how you simplify multiplication of binomials with two different signs (minus and plus) between the same type of terms.
We take an example with numbers:

(9 + 3)(9 - 3)
    9
+
3  
   
·
9
-
3  
 
 
      -9  
    - 27    
+
81 +27    
 
  81 + 0 - 9 = 72
We can even write this last calculation as powers and then it looks like this:
81
- 9
= 92 - 32

We try now to write the number with variables and replace the 9 with a and the 3 with b.
a
+
b
   
·
a
-
b
 
      - b2
    - ab  
    +ab  
+
a2    
 
  a2   - b2
Then, we can write the general formula.

In some math books, they cover the conjugate rule like this:

1. (a + b)(a - b)

4. (a + b)(a - b) = a - ab + ab - b

5. (a + b)(a - b) = a - b

Now we will try to see if the rule works.  We test it with the following algebraic example:

(2a + 3b)(2a - 3b)

We multiply together the expression in the parenthesis:

(2a + 3b)(2a - 3b) = 4a - 6ab + 6ab - 9b = 4a - 9b

Here we us the rule:

(2a + 3b)(2a - 3b) = 4a² - 9b²

We get the same answer.  As you can see it goes much quicker using the conjugate rule.