WEBVTT

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欢迎观看 这一节讲二次公式

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二次公式 听起来很复杂

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第一次见到它时

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你也许会说 何止听起来复杂

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完全就是复杂

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不过 随着课程推进

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你会发现并不难

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后面的课程中 我会告诉大家它是怎么来的

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大家已经知道

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如何因式分解二次方程

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比如 x2-x-6=0

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比如 x2-x-6=0

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可以分解为(x-3)(x+2)=0

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要么x-3=0 要么x+2=0

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x-3=0 或 x+2=0

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所以 x=3 或 -2

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我们作图看下

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函数为f(x)=x2-x-6

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纵轴是f(x)

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也许大家更习惯y轴表示

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这里不打紧

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横轴为x轴

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x2-x-6的图像大概像这样

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x2-x-6的图像大概像这样

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这是f(x)=-6

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图像大致是这样

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它将通过-6 因为x=0时

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f(x)=-6

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必然过这一点

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我还知道 f(x)=0在x轴上

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我还知道 f(x)=0在x轴上

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因为这是1

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这是0

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这是-1

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x轴与曲线相交处就是f(x)=0

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我们知道 这时 x要么等于3

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要么等于-2

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就是这个方程的解

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我们进行因式分解时

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并没有想几何意义

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这相当于将这样一个函数f(x)

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设为0

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然后问 该函数何时为0

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然后问 该函数何时为0

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在这两点为0

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这就是f(x)=0的地方

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这里我们所做的 是通过因式分解

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求出让f(x)=0的值

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也就是这两点

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下面讲一下数学词汇

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这些称为f(x)的"零点" 或者说"根"

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比如 f(x)=x2+4x+4

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求f(x)的零点或根

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等价于问

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f(x)与x轴的交点在哪

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交点处 f(x)=0 对吧

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想想我刚画的图像

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如果f(x)=0 则有

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0=x2+4x+4

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因式分解有(x+2)(x+2)

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于是 x=-2时 f(x)=0

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这是多余的 x=-2

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现在我们知道

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容易分解的方程如何求零点了

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下面来看一个不容易因式分解的情况

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下面来看一个不容易因式分解的情况

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f(x)=-10x2-9x+1

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如果除以10的话

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会得到分数

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因式分解起来有点费事

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因式分解这个二次多项式

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因式分解这个二次多项式

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看看这个何时为0

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看看这个何时为0

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-10x2-9x+1

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要求它何时为0

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这就可以用到二次公式这一工具

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下面需要大家简单记一些数学公式

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下面需要大家简单记一些数学公式

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二次方程的根是…

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假设二次方程是

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Ax2+Bx+C=0

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本例中 A=-10

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B=-9 C=1

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公式是 根x等于-B加减

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根号下(B2-4AC)

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整个除以2A

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看起来有点复杂 用多了之后

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会发现其实并没那么糟糕

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这个最好还是记住

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将公式用到刚才这个方程

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将公式用到刚才这个方程

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看看

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A是x2项系数

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A是x2项系数

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B是x项系数 C是常数项

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用到之前那个方程

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B是多少

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B是-9

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看这里

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B=-9 A=-10

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C=1

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B=-9 于是这是-(-9)

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加减根号下 -9的平方

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即81

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减4AC

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A是-10

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C则是1

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有点乱 但愿大家看得懂

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所有这些除以2A

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A=-10 所以是-20 化简

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负负得正 首先是+9

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加减 根号下 81

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这里有-4乘以-10

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这是-10

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有点看不清 抱歉 再乘以1

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-4×(-10)=40

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正40

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所有这些除以-20

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81+40=121

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于是9加减根号121 除以-20

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根号121=11

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写到这里

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但愿你们看得明白

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有(9±11)/(-20)

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(9+11)/(-20)=20/(-20)=-1

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(9+11)/(-20)=20/(-20)=-1

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-1是一个根

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由于这里是加减号

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所以另一个根是(9-11)/(-20)

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也就是-2/(-20)

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等于1/10

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这就是另一个根

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如果画图的话 我们会看到

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它同x轴的交点处

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或者说f(x)=0处 x=-1或1/10

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第二部分我会给出更多例子

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这一节但愿没让大家困惑

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第二部分再见

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