8.7 Conjugate Rule
The conjugate rule is a rule about how you expand, simplify and write multiplication for two binomials with different signs (minus and plus) in a general for – in other words, so that it applies for all numbers.
In basic algebra, chapter 4, you can learn how to do algebraic multiplication. You will need this knowledge in order to understand how the conjugate rule works. We take the following example:
(9 + 3)(9  3)




9 
+ 
3 




· 
9 
 
3 









9 




 27 



+ 
81 
+27 







81 
+ 0 
 9 
= 72 

We can even write the last calculation as a power and then it looks like this: 



= 91  9 =
=
9^{2 } 3^{2}




We take the first number in the square are subtract it with the second one squared. 

In books, the conjugate rule might be explained as follows:
1. (7 + 3)(7  3) 


4. (7 + 3)(7  3) = 49  21 + 21  9
5. (7 + 3)(7  3) = 49  9
6. (7 + 3)(7  3) = 49  9 = 40 

Now we can prove it this works. We test it using the following example:
(9 + 2)(9  2)
Here we multiply together the parenthesis.
(9 + 2)(9  2) = 81  18 + 18  4 = 77
Here we use the rule and take the first term squared and subtract the second term squared.
(9 + 2)(9 
2) = 9²  2² = 81 – 4 = 77
The rules works and we get the same answer! 

We can summarise the conjugate rule like this: 

The first term squared subtracted with the second term squared. 