# 和平方

（重定向自三數和平方

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

## 驗證

### 基本驗證

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

### 簡單驗證

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

### 幾何驗證

• ${\displaystyle a^{2}}$
• ${\displaystyle ab}$
• ${\displaystyle ab}$
• ${\displaystyle b^{2}}$

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

## 三數和平方

${\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ac\,\!}$

### 驗證

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

${\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ac}$
x) ${\displaystyle a}$  ${\displaystyle +b}$  ${\displaystyle +c}$
${\displaystyle a}$  ${\displaystyle a^{2}}$  ${\displaystyle +ab}$  ${\displaystyle +ac}$
${\displaystyle +b}$  ${\displaystyle +ab}$  ${\displaystyle +b^{2}}$  ${\displaystyle +bc}$
${\displaystyle +c}$  ${\displaystyle +ac}$  ${\displaystyle +bc}$  ${\displaystyle +c^{2}}$

${\displaystyle a^{2}+b^{2}+c^{2}+ab+ab+bc+bc+ac+ac}$
${\displaystyle =a^{2}+b^{2}+c^{2}+2ab+2bc+2ac}$