# 平方差

（重定向自平方差公式

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

${\displaystyle (a+b)}$${\displaystyle (a-b)}$的排列不是非常的重要，可隨意排放。 也可叫做和差積，（a+b）是和、（a-b）是差，兩個乘起來，是積。

## 驗證

### 主驗證

{\displaystyle {\begin{aligned}a^{2}-b^{2}&=a^{2}-0-b^{2}\\&=a^{2}-(ab-ba)-b^{2}\\&=a^{2}-ab+ba-b^{2}\\&=(a^{2}-ab)+(ba-b^{2})\\&=a(a-b)+b(a-b)\\&=(a-b)(a+b)\\\end{aligned}}}

### 方格驗證

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

### 幾何驗證

#### 方法一

• ${\displaystyle b(a-b)\,\!}$
• ${\displaystyle (a-b)^{2}\,\!}$ 是灰正方
• ${\displaystyle b(a-b)\,\!}$

${\displaystyle b(a-b)+(a-b)^{2}+b(a-b)\,\!}$
${\displaystyle =ab-b^{2}+a^{2}-2ab+b^{2}+ab-b^{2}\,\!}$
${\displaystyle =ab+ab-2ab-b^{2}+b^{2}+a^{2}-b^{2}\,\!}$
${\displaystyle =a^{2}-b^{2}\,\!}$
• 註：${\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}}$ 运用了差平方

#### 方法二

• ${\displaystyle a(a-b)\,\!}$ 大長方
• ${\displaystyle b(a-b)\,\!}$ 小長方

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

## 例子

### 例子一

${\displaystyle x^{2}-16\,\!}$

${\displaystyle =x^{2}-4^{2}\,\!}$
${\displaystyle =(x-4)(x+4)\,\!}$

### 例子二

${\displaystyle 16m^{2}-81n^{2}\,\!}$

${\displaystyle =(4m)^{2}-(9n)^{2}\,\!}$
${\displaystyle =(4m-9n)(4m+9n)\,\!}$

### 例子三

${\displaystyle 4y^{2}-36z^{2}\,\!}$

${\displaystyle =4(y^{2}-9z^{2})\,\!}$

${\displaystyle =4\left[y^{2}-(3z)^{2}\right]\,\!}$
${\displaystyle =4(y-3z)(y+3z)\,\!}$

### 例子四

${\displaystyle {\frac {1}{x^{4}}}-{\frac {13}{x^{2}}}+36}$

${\displaystyle =x^{-4}-13x^{-2}+36\,\!}$
${\displaystyle =(x^{-2})^{2}-13(x^{-2})+36\,\!}$

${\displaystyle =x^{-2}\times x^{-2}-9(x^{-2})-4(x^{-2})+9\times 4}$
${\displaystyle =(x^{-2}-4)(x^{-2}-9)\,\!}$

${\displaystyle =\left[(x^{-1})^{2}-2^{2}\right]\left[(x^{-1})^{2}-3^{2}\right]}$

${\displaystyle =(x^{-1}-2)(x^{-1}+2)(x^{-1}-3)(x^{-1}+3)\,\!}$

${\displaystyle =\left({\frac {1}{x}}-2\right)\left({\frac {1}{x}}+2\right)\left({\frac {1}{x}}-3\right)\left({\frac {1}{x}}+3\right)\,\!}$

## 運用

### 用平方差代替整數相乘

• ${\displaystyle 10\times 10=(10-0)(10+0)=10^{2}-0^{2}=100-0=100}$
• ${\displaystyle 7\times 13=(10-3)(10+3)=10^{2}-3^{2}=100-9=91}$
• ${\displaystyle 95\times 105=(100-5)(100+5)=100^{2}-5^{2}=10,000-25=9,975}$
• ${\displaystyle 99,994\times 100,006=(100,000-6)(100,000+6)=100,000^{2}-6^{2}=10,000,000,000-36=9,999,999,964}$

• ${\displaystyle 14^{2}-4^{2}=(14+4)(14-4)=18\times 10=180\,\!}$
• ${\displaystyle 125^{2}-25^{2}=(125+25)(125-25)=150\times 100=15,000\,\!}$
• ${\displaystyle 1,750^{2}-750^{2}=(1,750+750)(1,750-750)=2,500\times 1,000=25,000,000\,\!}$
• ${\displaystyle 14,205^{2}-4,205^{2}=(14,205+4,205)(14,205-4,205)=18,410\times 10,000=184,100,000\,\!}$

• ${\displaystyle 3263\times 3264\times \left({\frac {3264}{3263}}-{\frac {3265}{3264}}\right)}$

${\displaystyle =3263\times 3264\times {\frac {3264}{3263}}-3263\times 3264\times {\frac {3265}{3264}}}$

${\displaystyle =3264^{2}-3263\times 3265}$

${\displaystyle =3264^{2}-(3264-1)(3264+1)\,\!}$
${\displaystyle =3264^{2}-(3264^{2}-1^{2})\,\!}$
${\displaystyle =3264^{2}-3264^{2}+1\,\!}$
${\displaystyle =1\,\!}$

### 錯誤運用

 ${\displaystyle a^{2}-b^{2}=\left(a+b\right)\left(a-b\right)}$ ${\displaystyle a^{2}-b^{2}=(a-b)^{2}\,\!}$
• 註：${\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}}$  ，詳見差平方