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欢迎观看这一节视频课 这一节讲配平

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什么叫配平呢

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配平是解二次方程的一张方法

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所以在解一个二次方程前...

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我先把这个二次方程写下来

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然后我将说明如何配平

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然后再看其它例子

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以及为什么这叫作配平

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方程是x2+16x-57=0

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根据已学的知识

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我们可以使用因式分解

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看哪两个数加起来等于16

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且乘起来是-57

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稍微想想

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我们得到的并不一定是整数

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这一例虽然是整数

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但情况并非总是如此

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所以 因式分解只能用到

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确定能得到整数表达式的情况

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(x+整数)(x+整数)这种形式

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(x+整数)(x+整数)这种形式

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另一个方法是二次公式

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我们最终会看到

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二次公式其实是配方的快捷方式

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它其实是通过配方得到的

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那么配方是什么 怎么做呢

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首先看看 如何展开平方式

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首先看看 如何展开平方式

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做在下面这里

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(x+a)2是多少

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它等于x2+2ax+a2

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任何这种形式的

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都可以写成(x+某数)2

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如果能把这个方程

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写成(x+a)2=某数的形式

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就能直接开方求解

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配方所要做的也正是这些 我举例说明下

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配方所要做的也正是这些 我举例说明下

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例子更容易理解

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这个框起来 是需要记住的

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配方其实也就是这个

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让方程一侧得到这种形式

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另一侧只剩一个数

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然后两侧同时取平方根

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首先

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确认这不是一个完全平方式

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若是 将x项系数看作是2a

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a就是8 a2应该是64

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常数项显然不是64 不是完全平方式

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然后 我们可以

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在两侧同时加上57 以去掉57

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有x2+16x=57

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这是在两侧同时加上57

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然后 左侧需要加上什么

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才能得到(x+a)2这样的式子呢

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按照下面的规律 有x2

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+2ax 这个看成2ax

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这是2ax

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然后需要加上a2

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加a2 这个形式就有了

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但方程一侧进行运算 另一侧

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需要进行相同运算

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左侧加a2 右侧也要加a2

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这就是完全平方式的形式了

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但还需要知道a

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a是多少呢

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如果这个是2ax

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2a显然是16 所以a为8

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这只用观察法就能得出

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写出来的话

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也就是2ax=16x

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然后两侧同时除以2x a=16x/2x

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假设x不为0 则a=8

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a=8 表达式可以写成

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随便换种颜色 x2+16x+64

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a是8 a2也就是64

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等于57+64

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我的说明有些冗长

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其实这里到这里 也就是两侧同时加57

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57移到右侧

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然后同时加64

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加64是为了

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让左侧得到这个完全平方形式

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得到完全平方形式后

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重写出来是

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(x+a)2 这个形式

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而a是8 所以是(x+8)2

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等于57+64 也就是121

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这就非常好做了

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仍然是二次方程

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仍然是二次方程

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却不需要公式法或因式分解了

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可以直接两侧同时开方

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同时开方得到什么

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再随便换个颜色

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开方得到x+8=±根号121 正负号别忘了

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根号121等于多少 11

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到这里来 不管这个 这个只是草稿

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于是有x+8=±11

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两侧同时-8 有x=-8±11

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x可以是-8+11 也就是3

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确保我没做错

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x=-8±11

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对的 x可以是3

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或者-8-11 x也可以是-19

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看看说不说得通

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理论上 这个可以分解为

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(x-3)(x+19)=0

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因为这两个是方程的解

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这很正确

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-3×19=-57

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而x项系数-3+19=16

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这个题其实可以直接用分解的

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不过这并不明显

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因为19是个很奇怪的数 还不如用配方

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为什么叫配方呢

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因为需要得到这种形式 加64

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来配完整这个式子

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让左侧得到完全平方式

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再看个例子

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我将减少说明 快速完成

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这样可能看起来会更简单

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不过这个问题更麻烦一些

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6x2-7x-3=0

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当然 还是可以用因式分解 但我不喜欢

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x2项有系数时使用因式分解

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两侧同时除以6我也不喜欢

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这样到处都是分数

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通过观察法分解不好使

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可以用二次公式

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之后的视频中我会讲到

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其实我已经算是讲到了

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二次公式本质就是配方

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它只是一种快捷方式 将配方公式化

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还是配方吧

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这才是这一节的主题

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首先 两侧同时加上3

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得到6x2-7x=3

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有些老师喜欢不管这个-3

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直接配方

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我觉得先把3挪开

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算起来更清楚

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这个6我也不喜欢 它会把事情搞复杂

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我希望得到(x+a)2的形式 不希望x前

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还有系数

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所以 两侧同时除以6

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得到x2-7/6x=3/6=1/2

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这可以一步就位

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直接把6一起除过去

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下面开始配方

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x2-7/6x+某数等于1/2 留点空间

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左侧需要加上某数

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得到完全平方式

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怎么做呢

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观察系数

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这里不是7/6 而是-7/6

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取它的1/2 然后平方

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算算

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(x+a)2=x2+2ax+a2

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这是需要记住的

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也是所有配方的基础

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我刚说

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这一项是x项系数1/2的平方

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为什么

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因为将两个式子的样式进行匹配

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我们知道 a就是1/2的x项系数

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-7/6的1/2是-7/12

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写出来的话 这里a=-7/12

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也就是x项系数乘以1/2

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两侧需要加多少 才能完成配方

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需要加(-7/12)2 也就是49/144

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两侧同时加上49/144

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左侧如何化简呢

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我们知道这是完全平方式

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我们知道a是-7/12

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所以方程左侧是

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(x+a)2 而a是负数

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将其乘开必然得到这个

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右边需要通分 公分母是144

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72+49=121 所以是121/144

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所以(x-7/12)2=121/144

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然后

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两侧同时开平方

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找点空位置 换绿色

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把这里框出来

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我们有x-7/12=±根号(121/144)

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也就是±11/12

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根号121是11 根号144是12

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然后两侧同时加上7/12

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有x=7/12±11/12

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也就是(7±11)/12

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有两种可能 7+11=18 18/12

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x=18/12 即3/2

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或7-11 有-4/12

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即x=-1/3 搞定 这就是配方

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但愿大家能够理解

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证明二次公式就需要配方

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只是要配方的不再是具体数例

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而是抽象方程Ax2+Bx+C=0

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对Ax2+Bx+C=0进行配方

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最后得到二次公式

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之前视频我好像讲过

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如果我没讲过 之后会讲

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好了 下次再会

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网易公开课官方微博 http://t.163.com/163open

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