5.3 Quadratic Expansion with Subtraction
In chapter 8.6 in the section Operators and Calculation Rules, we covered quadratic multiplication. Here we split this up into two pages, one for addition and one for subtraction.
We begin with the same example that we covered in chapter 8.6 in Operators and Calculation Rules: 


In chapter 4, you learned how to do algebraic multiplication, and you will with this knowledge be able to learn the rules for quadratic expansion for subtraction. 



10 
3 




· 
10 
3 









9 




30 



+ 
100 
30 







100 
60 
+ 9 
= 49 

We can even write the last calculation with power and it would then look like this. 
100 
60 
+ 9 

10^{2} 
 2(10 · 3) 
+ 3^{2} 



We begin with 7^{2}, which we wrote as a subtraction, (10  3)^{2}, and got that it becomes 10^{2}  2(10 · 3) + 3^{2} = 49.
We try now to write the number with variables and replace the 10 with a and the 3 with b. 



a 
b 



· 
a 
b 







b^{2} 



ab 




ab 


+ 
a^{2} 






a^{2} 
2ab 
+ b^{2} 

We can then write the formula: 

With this we have the “general formula” for the rule for subtraction. It is called the second rule for quadratic expansion. 

In some math books they cover quadratic expansion for subtraction like this:
1. (a  b)^{2} = (a – b)(a – b) 


4. (a  b)(a  b) = a²  ab  ba + b²
5. (a  b)² = a²  2ab + b² 