# Basic Algebra/Solving Equations/Formulas

## Basic Formulas

1. ${\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}$
2. ${\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}}$
3. ${\displaystyle a^{2}-b^{2}=(a+b)(a-b)}$
4. ${\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}}$
5. ${\displaystyle (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}}$
6. ${\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})}$
7. ${\displaystyle a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})}$
8. ${\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca}$
9. ${\displaystyle a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)}$

### Derivation (1)

${\displaystyle (a+b)^{2}}$
${\displaystyle =(a+b)(a+b)}$
${\displaystyle =a(a+b)+b(a+b)}$
${\displaystyle =a^{2}+ab+ab+b^{2}}$
${\displaystyle =a^{2}+2ab+b^{2}}$

### Derivation (2)

${\displaystyle (a-b)^{2}}$
${\displaystyle =(a-b)(a-b)}$
${\displaystyle =a(a-b)-b(a-b)}$
${\displaystyle =a^{2}-ab-ab+b^{2}}$
${\displaystyle =a^{2}-2ab+b^{2}}$

### Derivation (3)

${\displaystyle a^{2}-b^{2}}$
${\displaystyle =a^{2}+ab-ab-b^{2}}$
${\displaystyle =a(a+b)-b(a+b)}$
${\displaystyle =(a+b)(a-b)}$

### Derivation (4)

${\displaystyle (a+b)^{3}}$
${\displaystyle =(a+b)\times (a+b)^{2}}$
${\displaystyle =(a+b)\left(a^{2}+2ab+b^{2}\right)}$
${\displaystyle =a\left(a^{2}+2ab+b^{2}\right)+b\left(a^{2}+2ab+b^{2}\right)}$
${\displaystyle =a^{3}+2a^{2}b+ab^{2}+a^{2}b+2ab^{2}+b^{3}}$
${\displaystyle =a^{3}+3a^{2}b+3ab^{2}+b^{3}}$

### Derivation (5)

${\displaystyle (a-b)^{3}}$
${\displaystyle =(a-b)\times (a-b)^{2}}$
${\displaystyle =(a-b)\left(a^{2}-2ab+b^{2}\right)}$
${\displaystyle =a\left(a^{2}-2ab+b^{2}\right)-b\left(a^{2}-2ab+b^{2}\right)}$
${\displaystyle =a^{3}-2a^{2}b+ab^{2}-a^{2}b+2ab^{2}-b^{3}}$
${\displaystyle =a^{3}-3a^{2}b+3ab^{2}-b^{3}}$