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# 分解二次表达式：完全平方

## 介绍：分解完全平方三项式

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

$\begin{array}{rl}\left(x+5{\right)}^{2}& ={x}^{2}+2\left(x\right)\left(5\right)+\left(5{\right)}^{2}\\ \\ & ={x}^{2}+10x+25\end{array}$

$\begin{array}{rl}{x}^{2}+10x+25& ={x}^{2}+2\left(x\right)\left(5\right)+\left(5{\right)}^{2}\\ \\ & =\left(x+5{\right)}^{2}\end{array}$

## 例题 1: 因式分解 ${x}^{2}+8x+16$‍

$\begin{array}{rl}{x}^{2}+8x+16& =\left(x{\right)}^{2}+2\left(x\right)\left(4\right)+\left(4{\right)}^{2}\\ \\ & =\left(x+4{\right)}^{2}\end{array}$

$\begin{array}{rl}\left(x+4{\right)}^{2}& =\left(x{\right)}^{2}+2\left(x\right)\left(4\right)+\left(4{\right)}^{2}\\ \\ & ={x}^{2}+8x+16\end{array}$

### 看看你对知识掌握得如何

1) 因式分解 ${x}^{2}+6x+9$.

2) 因式分解 ${x}^{2}-6x+9$.

3) 因式分解 ${x}^{2}+14x+49$.

## 例题 2: 因式分解 $4{x}^{2}+12x+9$‍

$\begin{array}{rl}4{x}^{2}+12x+9& =\left(2x{\right)}^{2}+2\left(2x\right)\left(3\right)+\left(3{\right)}^{2}\\ \\ & =\left(2x+3{\right)}^{2}\end{array}$

### 检查你对本课的理解

4) 因式分解 $9{x}^{2}+30x+25$.

5) 因式分解 $4{x}^{2}-20x+25$.

## 挑战题

6*) 因式分解 ${x}^{4}+2{x}^{2}+1$.

7*) 因式分解 $9{x}^{2}+24xy+16{y}^{2}$.

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