8.6 Quadratic Expansion
Quadratic rules for addition
Now we are going to learn the quadratic rules for addition and we begin with an example: 


We can illustrate this in pictures, and now can see if we get the same results: 


We calculate each part by itself: 



3 · 3 

2(3 · 7) 

+ 
7 · 7 

3^{2} 
+ 
2(3 · 7) 

+ 
7^{2} 

9 
+ 
42 

+ 
49 
= 100 


We began with 10^{2}. The base 10 is divided into two terms, (3 + 7)^{2}, and we see that it becomes:
10^{2} = (3 + 7)^{2} = 3^{2} + 2(3 · 7) +
7^{2} = 100. 

In certain books, you might see the quadratic rules for addition explained as follows:
1. (7 + 3)² = (7 + 3)(7 + 3) 


4. (7 + 3)(7 + 3) = 49 + 21 + 21 + 9
5. (7 + 3)² = 49 + 2 · 21
+ 9
6. (7 + 3)² = 49 + 42 + 9 = 100 

Quadratic rules for subtraction
Now we continue with quadratic expansion for subtraction and we begin even here with an example: 


In the section, Algebra, in chapter 4, you can learn how to do algebraic multiplication. This knowledge is helpful to use when learning quadratic rules for subtraction. Algebraic multiplication looks as follows: 



10 
3 




· 
10 
3 










9 




30 



+ 
100 
30 








100 
60 
+ 9 
= 49 

We can even write the last calculation by using powers, and then see that it looks like this: 
100 
60 
+ 9 

10^{2} 
 2(10 · 3) 
+
3^{2} 



We began with 7^{2}. We divided the base 10 into two terms, (10 
3)^{2}, and figured out that it becomes:
7^{2} = (10  3)^{2 }= 10^{2}  2(10 · 3) + 3^{2} = 49. 

In certain books, quadratic expansion for subtraction might look like this:
1. (7  3)² = (7  3)(7  3) 


4. (7  3)(7  3) = 49  21  21 + 9
5. (7  3)² = 49  2 · 21 + 9
6. (7  3)² = 49  42 + 9 = 16 

We can summarise quadratic expansion as follows:
Quadratic expansion for addition: 

Quadratic expansion for subtraction: 
