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歡迎觀看 這一節講二次公式

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二次公式 聽起來很複雜

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第一次見到它時

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你也許會說 何止聽起來複雜

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完全就是複雜

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不過 隨著課程推進

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你會發現並不難

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後面的課程中 我會告訴大家它是怎麽來的

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大家已經知道

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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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可以分解爲(x-3)(x+2)=0

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要麽x-3=0 要麽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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我們作圖看下

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函數爲f(x)=x2-x-6

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縱軸是f(x)

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也許大家更習慣y軸表示

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這裡不打緊

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橫軸爲x軸

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x2-x-6的圖像大概像這樣

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x2-x-6的圖像大概像這樣

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這是f(x)=-6

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圖像大致是這樣

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它將通過-6 因爲x=0時

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f(x)=-6

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必然過這一點

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我還知道 f(x)=0在x軸上

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我還知道 f(x)=0在x軸上

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因爲這是1

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這是0

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這是-1

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x軸與曲線相交處就是f(x)=0

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我們知道 這時 x要麽等於3

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要麽等於-2

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就是這個方程的解

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我們進行因式分解時

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並沒有想幾何意義

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這相當於將這樣一個函數f(x)

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設爲0

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然後問 該函數何時爲0

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然後問 該函數何時爲0

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在這兩點爲0

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這就是f(x)=0的地方

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這裡我們所做的 是通過因式分解

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求出讓f(x)=0的值

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也就是這兩點

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下面講一下數學詞彙

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這些稱爲f(x)的"零點" 或者說"根"

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比如 f(x)=x2+4x+4

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求f(x)的零點或根

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等價於問

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f(x)與x軸的交點在哪

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交點處 f(x)=0 對吧

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想想我剛畫的圖像

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如果f(x)=0 則有

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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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於是 x=-2時 f(x)=0

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這是多余的 x=-2

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現在我們知道

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容易分解的方程如何求零點了

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下面來看一個不容易因式分解的情況

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下面來看一個不容易因式分解的情況

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

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如果除以10的話

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會得到分數

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因式分解起來有點費事

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因式分解這個二次多項式

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因式分解這個二次多項式

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看看這個何時爲0

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看看這個何時爲0

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-10x2-9x+1

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要求它何時爲0

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這就可以用到二次公式這一工具

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下面需要大家簡單記一些數學公式

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下面需要大家簡單記一些數學公式

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二次方程的根是…

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假設二次方程是

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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等於-B加減

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根號下(B2-4AC)

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整個除以2A

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看起來有點複雜 用多了之後

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會發現其實並沒那麽糟糕

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這個最好還是記住

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將公式用到剛才這個方程

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將公式用到剛才這個方程

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看看

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A是x2項係數

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A是x2項係數

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B是x項係數 C是常數項

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用到之前那個方程

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B是多少

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B是-9

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看這裡

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B=-9 A=-10

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C=1

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B=-9 於是這是-(-9)

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加減根號下 -9的平方

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即81

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減4AC

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A是-10

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C則是1

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有點亂 但願大家看得懂

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所有這些除以2A

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A=-10 所以是-20 化簡

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負負得正 首先是+9

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加減 根號下 81

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這裡有-4乘以-10

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這是-10

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有點看不清 抱歉 再乘以1

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-4×(-10)=40

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正40

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所有這些除以-20

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81+40=121

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於是9加減根號121 除以-20

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根號121=11

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寫到這裡

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但願你們看得明白

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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是一個根

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由於這裡是加減號

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所以另一個根是(9-11)/(-20)

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也就是-2/(-20)

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等於1/10

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這就是另一個根

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如果畫圖的話 我們會看到

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它同x軸的交點處

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或者說f(x)=0處 x=-1或1/10

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第二部分我會給出更多例子

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這一節但願沒讓大家困惑

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第二部分再見

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