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# 分解二次表达式：平方差

## 简介:平方差公式

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

$\begin{array}{r}{x}^{2}-{2}^{2}=\left(x+2\right)\left(x-2\right)\end{array}$

$\begin{array}{rl}\left(x+2\right)\left(x-2\right)& =x\left(x-2\right)+2\left(x-2\right)\\ \\ & ={x}^{2}-2x+2x-4\\ \\ & ={x}^{2}-4\end{array}$

## 例题 1：分解 ${x}^{2}-16$‍  .

${x}^{2}-16=\left(x{\right)}^{2}-\left(4{\right)}^{2}$

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

$\left(x{\right)}^{2}-\left(4{\right)}^{2}=\left(x+4\right)\left(x-4\right)$

### 看看你对知识掌握得如何

1)因式分解${x}^{2}-25$.

2) 因式分解 ${x}^{2}-100$.

### 反思题

3) 我们能否利用平方差公式来分解 ${x}^{2}+25$?

## 例题 2：因式分解 $4{x}^{2}-9$‍

$\begin{array}{rl}4{x}^{2}-9& =\left(2x{\right)}^{2}-\left(3{\right)}^{2}\\ \\ & =\left(2x+3\right)\left(2x-3\right)\end{array}$

### 看看你对知识掌握得如何

4) 因式分解 $25{x}^{2}-4$.

5) 因式分解 $64{x}^{2}-81$.

6) 因式分解 $36{x}^{2}-1$.

## 挑战题

7*) 因式分解 ${x}^{4}-9$.

8*) 因式分解 $4{x}^{2}-49{y}^{2}$.