Quadratic Equation

ting

If a, b and c are real numbers and not all equal, prove that the quadratic equation
(c-a)x^2-2(a-b)x+(b-c)=0 has unequal real roots.

計不到
開始對自己適不適合讀理科產生懷疑

kazuhiko

discriminant
= [2(a-b)]^2 - 4(c-a)(b-c)
= 4(a-b)^2 - 4(c-a)(b-c)
= 4[ a^2 - 2ab + b^2 - (bc - c^2 - ab + ac)]
= 4( a^2 + b^2 + c^2 - ab - bc - ac )
= 2[(a-b)^2 + (b-c)^2 + (c-a)^2] > 0  (a, b and c are real numbers and not all equal)

Therefore, the quadratic equation (c-a)x^2-2(a-b)x+(b-c)=0 has unequal real roots

[ 本帖最後由 kazuhiko 於 24/9/2006 21:39 編輯 ]

Royal

那是附加數抑或是普通數學?
我沒有修附加數, 但有學類似的東東,也計不到xd

ting

amths來的
計到4( a^2 + b^2 + c^2 - ab - bc - ac )就卡住了
想不到開學第一堂amath(沒硬性把amath和maths分開)已有問題
灰到爆開

[ 本帖最後由 (ABC)ting 於 24/9/2006 12:01 編輯 ]

魔術師Chikorita

原帖由 kazuhiko 於 23/9/2006 05:40 PM 發表
determinant
= ^2 - 4(c-a)(b-c)
= 4(a-b)^2 - 4(c-a)(b-c)
= 4
= 4( a^2 + b^2 + c^2 - ab - bc - ac )
= 2 > 0  (a, b and c are real numbers and not all equal)

Therefore, the quadratic equat ...

a=1, b=2, c=3的時候已經錯了......

ting

原帖由 魔術師Chikorita 於 24/9/2006 12:03 發表

a=1, b=2, c=3的時候已經錯了......

wt do u "up"?
who tell u a=1 , b=2, c=3.
You assume that?
a, b, c are only for discriminant.
They are constant terms.
When u substitute a=1 b=2 c=3, x is still a variable.
How could u know it is wrong?
I think u can only assume 2 of them in order to find the range of the last one

[ 本帖最後由 (ABC)ting 於 24/9/2006 12:29 編輯 ]

魔術師Chikorita

原帖由 (ABC)ting 於 24/9/2006 12:17 PM 發表

wt do u "up"?
who tell u a=1 , b=2, c=3.
You assume that?
a, b, c are only for discriminant.
They are constant terms.
When u substitute a=1 b=2 c=3, x is still a variable.
How ...

又學多一個東東了

流映

唔需要灰心…呢條唔識是正常的

Royal

原帖由 kazuhiko 於 23/9/2006 17:40 發表
determinant
= ^2 - 4(c-a)(b-c)
= 4(a-b)^2 - 4(c-a)(b-c)
= 4
= 4( a^2 + b^2 + c^2 - ab - bc - ac )
= 2 > 0  (a, b and c are real numbers and not all equal)

Therefore, the quadratic equat ...

係唔係(a-b)^2一定會係正數架?

**估唔到amaths真係咁難...好彩冇修

ting

原帖由 Royal 於 24/9/2006 12:59 發表

係唔係(a-b)^2一定會係正數架?

**估唔到amaths真係咁難...好彩冇修

yes
square must be positive